Senate backers of a bill the White House fears could dismantle a potential nuclear deal with Iran are closing in on a veto-proof threshold of support. The Senate Foreign Relations Committee is expected to approve the bill next Tuesday and hold a full Senate vote soon after. The measure provides a skeptical Congress with the opportunity to review the deal and prevents the president from waiving congressional sanctions on Iran during the review period. The White House has engaged in an extensive lobbying campaign on the Hill and with constituent groups concerned about Iran ever since a framework agreement was announced on April 2. The U.S. and the other world powers negotiating with Tehran have until June 30 to hammer out the final details to seal a deal. READ: Schumer: Let Congress decide on Iran deal But so far, the administration hasn’t been able to win over all the Senate Democrats, many of whom believe strongly that Congress has a constitutional obligation to weigh in on a major nuclear agreement with a long-time enemy of the United States. The bill already has nine Democratic co-sponsors and a handful of other Democrats have either expressed support or remain open to backing the bill. When combined with the Senate Republicans and one independent who support the legislation, that leaves backers just four shy of the 67 needed to sustain the veto that Obama has promised. Schumer’s support key The bill’s proponents got an important boost with the recent news that New York Sen. Chuck Schumer, a co-sponsor, was in line to become the Senate’s Democratic leader in 2017. The expected ascension has put his support in the spotlight since the popular Schumer could clear the way for more Democrats to back the bill. Schumer, a staunch supporter of Israel, also represents a large Jewish constituency, many of whom have expressed concerns about the framework Iran deal. The looming Iran agreement is vehemently opposed by Israeli Prime Minister Benjamin Netanyahu, and the GOP believes the split among Jewish voters on the issue will allow it to make inroads within the demographic group, which usually votes heavily Democratic. Schumer’s bucking the White House may signal to Jewish voters that he’ll be fighting to get the best Iran deal possible. What’s not clear is whether Democrats who vote for the bill, co-authored by Foreign Relations Committee Chairman Bob Corker and one of the committee’s senior Democrats, Robert Menendez of New Jersey, would also vote to override the veto of a Democratic president. CNN asked Schumer last month whether he would defy the President with a vote to override, but he refused to answer. “I’m not going to deal in hypotheticals,” Schumer responded. Some Democrats who are open to the bill have been working to make it more palatable to Obama, who opposes Congress’s effort to block him from lifting congressionally mandated sanctions against Iran during lawmakers’ 60-day review period. Since there is little chance that the Republican-controlled Congress will vote to lift the sanctions outright, Obama could meet Iranian demands for sanctions relief by unilaterally waiving them during his tenure – if the Corker-Menendez bill doesn’t pass and rule that out during the 60-day period. Sanctions relief is the key impetus for Iran to make the concessions on its nuclear program outlined in the framework agreement. Tehran has shown its own hesitance about the terms, with Iranian President Hassan Rouhani warning on Thursday that his country would “not sign any deal unless all economic sanctions are lifted on the same day of the implementation of the deal.” That is contrary to the framework provisions spelled out by the U.S., as well as the administration’s statement that it would take at least six months to undo the sanctions after completing an agreement. The Obama administration has warned that congressional action on the bill, known as the Iran Nuclear Agreement Review Act of 2015, could disrupt the fragile negotiations still underway. READ: Iran: No signing final nuclear deal unless economic sanctions are lifted The White House has said it supports Congress expressing itself on the deal but has not specified what form of expression would be acceptable. Meanwhile, it has made its concerns about the Corker-Menendez legislation quite clear in discussions with Democrats on Capitol Hill. Maryland Democratic Sen. Ben Cardin, who replaced Menendez as the top Democrat on the Foreign Relations Committee after Menendez was indicted on corruption charges, has been among the bill’s potential Democratic supporters trying to find a way to bridge the gap with the White House. Democrats try to soften language He has been in regular talks with Obama aides and is expected to offer amendments at Tuesday’s session. As the top Democrat on the committee, his decision in support or opposition could sway others in his party. Meanwhile, Delaware Sen. Chris Coons plans to propose an amendment that would remove what the White House says is a key concern: language that requires Iran to renounce terrorism as a prerequisite for a deal, which the administration sees as unrealistic and not focused on the nuclear issue. It’s unclear if Republicans would go along with that change. Other Democrats are weighing amendments to assuage specific White House concerns, according to aides. One expected amendment would shorten the Congressional review time from 60 days to 30. Already the bill reflects input from key Democrats toning down its language, but that has not satisfied the administration. “They don’t like the bill. They don’t like me being on the bill,” Virginia Sen. Tim Kaine told MSNBC Wednesday about his discussions with the White House. Kaine originally opposed the bill but signed on after securing changes from Corker that focused the bill only on the sanctions Congress approved, not those put in place by the United Nations or by the President through executive order. Kaine said he’s pointed out to the administration that it’s negotiating over sanctions that are under Congress’s jurisdiction. “If you’re going to negotiate over a congressional act, Congress is going to be involved,” Kaine said of the argument he’s been making to the White House. Though significant numbers of Democrats favor the bill, it has caused infighting in the caucus because the majority of Senate Democrats oppose it and want to give the President room to secure a deal with Iran. California Sen. Barbara Boxer, a member of the Foreign Relations Committee and a liberal ally of the President, is pushing for Corker to delay action on the bill, though it’s highly unlikely he will. “To force Congress to weigh in now on the Iran nuclear talks before a final deal has been completed would be a reckless rush to judgment,” Boxer said in a letter to Corker. “It would undermine negotiations at a critical moment and could derail a once-in-a-lifetime opportunity to deal with this looming threat.” READ: Tom Cotton: Bombing Iran would take ‘several days’ Boxer also wants a vote on an amendment that would abolish the Corker-Menendez bill and replace it with a requirement that the administration regularly report to Congress on whether Iran is violating the terms of an agreement. If they do, the proposed amendment sets up expedited procedures for Congress to re-impose sanctions against Iran. Boxer is backed by other liberals like Dianne Feinstein of California, Sherrod Brown of Ohio, and Al Franken of Minnesota. What Boxer’s proposal does not do is set up an up-or-down vote on the merits of an agreement with Iran, something most Republicans are anxious to do. They argue that a deal of this magnitude requires giving Congress a say. “This bill is not about the deal. This bill is about the review process. This bill passes no judgment on any final agreement that might be reached and whether it’s good or bad for the country,” said a congressional Republican aide familiar with the legislation but not authorized to speak on the record. Supporting the bill and the deal Supporters of Corker-Menendez note that it’s very possible that, when the vote that the bill requires comes, some of them will vote to approve the Iran deal if the multilateral talks secure a strong agreement. However, opponents, including the White House, have strong doubts any Republicans will back a final deal and believe some in the GOP are using the Corker-Menendez bill to disrupt the talks and prevent a final deal from being reached. “We have problems with this legislation in its own right, but our principal focus right now is, let’s give time and space to the negotiators to see if they get this done,” Ben Rhodes, the White House Deputy National Security Advisor, said in an interview this week with CNN. “Do not take action over the course of the next couple of months that could jeopardize those negotiations by backing the Iranians and the rest of the world away from the table,” he warned. President Barack Obama reached out personally to Corker on the issue Wednesday, the first time the men have spoken since the framework agreement was reached last week. Aides refused to divulge the specifics of what they discussed. But Obama added his own note of caution on the negotiations during a trip to Jamaica Thursday. “This is not done until it’s done, and the next two to three months of negotiations are going to be absolutely critical,” he said, describing the goal as “an agreement that gives us confidence and gives the world confidence that Iran in fact is not pursuing a nuclear weapon.” Despite the administration’s objections, Senate Republicans are unlikely to hold back on the bill until the end of June. And the GOP-led House will quickly take up the Corker-Menendez bill once it clears the Senate, according to a House Republican leadership aide. In the meantime, lawmakers will have multiple opportunities to get classified briefings from the administration on the framework agreement. Previous Iran sanctions votes have cleared the House on overwhelmingly bipartisan votes. But it’s uncertain if the Corker-Menéndez legislation will get as many Democratic votes after House Minority Leader Nancy Pelosi came out against it on Wednesday. “Senator Corker’s legislation undermines these international negotiations and represents in unnecessary hurdle to achieving a strong, final agreement,” Pelosi said in a statement. “In the weeks ahead, we must give this diplomatic framework room to succeed so that we can judge a June 30th agreement on its merits.” If the Senate clears the veto-proof hurdle, the Corker measure will certainly pass with significant bipartisan support in the House. But if Senate Democrats are unable to get the kinds of changes to the bill that the administration wants, it will fall to Pelosi to cajole enough Democrats to vote against the measure so that the House can’t override the expected presidential veto.