Washington (CNN) -- If you're looking for a sign that this week is shaping up as a critical test of President Barack Obama's presidency, look no further than his schedule on Tuesday.
In the morning, he'll make his fifth address to the United Nations General Assembly -- this time, with major diplomatic opportunities emerging in Syria and Iran. Later, his domestic agenda takes over when he turns again to former President Bill Clinton to explain Obamacare a week ahead of a critical date in its implementation.
Obama's health care law is the source of contention in Washington, where Republicans voted to defund it -- for the 42nd time -- in return for keeping the government running. That's not likely to go anywhere with Democrats who control the Senate -- meaning the country is still headed toward a government shutdown at the beginning of next week.
It all makes for a lengthy presidential to-do list, full of tasks that must be completed either within windows of opportunity or before hard-set deadlines pass.
Diplomatic openings abound
Obama arrives at this year's United Nations General Assembly amid diplomatic wrangling on multiple fronts, all of which could either bolster or damage his international legacy. A new opening to negotiate with Iran on its nuclear program, a framework for ridding Syria of its chemical weapons, and rebooted peace talks between Israelis and Palestinians all await Obama at his fifth General Assembly as president.
There is "a lot of diplomatic activity taking place as we go into the United Nations General Assembly, and we want to take advantage of those opportunities," Deputy National Security Adviser Ben Rhodes told reporters Monday as Obama flew to New York.
Speculation that Obama will participate in some exchange with newly elected Iranian President Hassan Rouhani was ignited as the newly elected Iranian president began a concerted program of outreach to the West, signaling he was ready to negotiate an end to Iran's nuclear program. Apparently sprung from economic sanctions tightening around Iran, the outreach from Rouhani included an opinion piece in the Washington Post and interviews scheduled with American television networks this week, including CNN.
White House officials, who say Iran's actions speak louder than Rouhani's words, say they haven't yet scheduled a meeting between the two leaders, though they also haven't ruled out direct diplomacy between Obama and his counterpart.
"We're always open to diplomacy, if we believe it can advance our objectives," Rhodes said during a conference call Friday.
The U.N. General Assembly assumes extra urgency this year as the body faces questions about the use of chemical weapons in Syria. Action through the U.N. Security Council had been stalled by Russia until a deal was struck earlier this month that would require Syrian President Bashar al-Assad surrender his stockpile of chemical weapons to international control.
The plan, however, must still be sanctioned as a U.N. resolution, and a disagreement between the U.S. and Russia has emerged concerning the potential use of force if Syria doesn't comply with the timeline for surrendering its weapons.
In his speech to the U.N. On Tuesday, Obama will press his case for global action on Syria, reportedly including renewed calls to remove al-Assad from power.
In New York, he'll meet with the presidents of Nigeria and Lebanon, as well as Palestinian Authority leader Mahmoud Abbas, from whom he'll receive an update on Mideast peace negotiations that began again this summer under Secretary of State John Kerry. Israeli Prime Minister Benjamin Netanyahu is slated to visit the White House at the end of September.
Big battle in Washington
As Obama looks to further widen his diplomatic openings in New York, no signs have emerged yet that Congress and the White House will find a way to avert a month-end government shutdown. The two sides seem no closer to a solution than they were last week, when the U.S. House voted mostly along party lines for a measure that would keep the government running while stripping the Affordable Care Act of its funding.
The measure isn't going anywhere in the Democrat-controlled Senate, and Obama has vowed to reject it. And he continued his offensive against Republicans over the shutdown.
"This is an interesting thing to ponder, that your top agenda is making sure 20 million people don't have health insurance," Obama said Saturday at a dinner for the Congressional Black Caucus Foundation. "And you'd be willing to shut down the government and potentially default for the first time in United States history because it bothers you so much that we're actually going to make sure that everybody has affordable health care. Let me say as clearly as I can: It is not going to happen."
With the House bill headed nowhere, lawmakers and Obama have just a week to bang out a short-term deal that would keep the government funded.
Obama and House Speaker John Boehner spoke by telephone Friday evening, the only evidence so far the two sides are talking. White House press secretary Jay Carney told reporters on Monday the president would "likely" meet with congressional leaders to discuss the fiscal deadline, though no meetings are on the schedule yet.
Even if the White House and Congress can keep the government running, they face a second deadline in mid-October on raising the debt ceiling, an issue on which Obama has said he won't negotiate with Republicans. That's enraged some GOP lawmakers -- Boehner wondered aloud last week why Obama was negotiating with Russian President Vladimir Putin, but not Republicans.
The latest round of the recurring budget battle has left little space for Congress to debate what could be a signature second-term achievement for Obama, a total overhaul of the nation's immigration system. After a bipartisan vote in the Senate in June, progress hit a wall in the House, where Republicans are wary of voting for anything that could be described as "amnesty."
That leaves the Affordable Care Act as the landmark law of Obama's presidency -- and even as Republicans attempt to dismantle it, Obamacare faces a critical test in a week that has the president making efforts to educate confused Americans about the law.
Faced with a confused public and ad campaigns encouraging people to shun the new insurance marketplaces, Obama and his allies are hurriedly working to convince the millions of Americans without health insurance to sign up for government-sponsored exchanges by October 1.
They're a key piece of Obama's law, but if not enough young, healthy Americans enroll in the marketplace, costs will be higher for everyone who participates. Republicans and their allies -- hopeful that higher rates will convince more people the law should go -- have been working to dissuade Americans from signing on, including in a pair of Web videos last week that featured a demonic Uncle Sam performing medical exams.
"Don't let the government play doctor. Opt out of Obamacare," conclude the ads, which were backed by a libertarian group funded by the conservative Koch Brothers.
Obama will begin another push to explain the law amid his efforts at global diplomacy in New York, sitting down with his Democratic predecessor, former President Bill Clinton, in a public discussion of the law on Tuesday evening. Clinton previously delivered remarks defending the law at his presidential library in August.
Coming a week ahead of the exchanges' official opening date, a White House official describes the event as "part of a ramped up public education effort to reach Americans who want to sign up for new affordable options in the health insurance Marketplace."
That effort -- which continues Thursday when Obama touts his law in Maryland -- comes as polls show Americans, particularly those who Obama is trying to reach, don't understand the law. In a Wall Street Journal poll earlier this month, 76% of uninsured respondents said they didn't understand the Affordable Care Act or how it might affect them.
CNN Senior White House Correspondent Jim Acosta contributed to this report.