The committee had set a Monday deadline for the agency to provide the evidence, a source familiar with the matter has told CNN -- a deadline that did not appear to apply to the Senate Intelligence Committee's investigation.
Senate panel Chairman Richard Burr told CNN, "I've talked to all the appropriate people," adding later that he has received "sufficient" responses about the wiretapping claim.
"I would say from our conversations, what we've gotten are sufficient answers," the North Carolina Republican said.
Burr would not say if there was any evidence backing Trump's claim.
The source told CNN over the weekend that the House committee had sent letters throughout the intelligence community this week to obtain records related to Russia, which the Justice Department responded to Monday evening.
"This afternoon, the Department of Justice placed calls to representatives of the chairman and ranking member of the United States House Permanent Select Committee on Intelligence to ask for additional time to review the request in compliance with the governing legal authorities and to determine what if any responsive documents may exist," Justice Department spokeswoman Sarah Isgur Flores said in a statement.
A spokesman for Rep. Devin Nunes, chairman of the House Intelligence Committee, suggested the panel may subpoena the administration if their questions go unanswered.
"The Department of Justice has asked for more time to comply with the House Intelligence Committee's request for information related to possible surveillance of Donald Trump or his associates during the election campaign," Nunes spokesman Jack Langer said in a statement. "We have asked the department to provide us this information before the committee's open hearing scheduled for March 20. If the committee does not receive a response by then, the committee will ask for this information during the March 20 hearing and may resort to a compulsory process if our questions continue to go unanswered."
In the upper chamber, senators on the intelligence panel say they've asked officials to provide evidence of Trump's wiretapping claims. They suggest there wasn't a Monday deadline -- as the House committee has requested.
Asked if he has imposed a deadline the way the House panel did, Burr said: "The House operates differently than I do."
Burr also said it's possible former Trump associates could be hauled before his committee, adding that Trump confidante Roger Stone's contacts
with "Guccifer 2.0" is "part of an ongoing investigation." The Guccifer 2.0 persona, who claimed credit for hacking the Democratic National Committee's emails, was identified by the US intelligence community as a front for Russian intelligence.
Meanwhile, Republican Sen. Lindsey Graham, who also made a request to the Justice Department alongside fellow Judiciary Committee member Sheldon Whitehouse, D-Rhode Island, said he hasn't heard back from the agency.
"I'm getting very ill-tempered over this," the South Carolina senator told reporters. "I expect them to take some time, but Judiciary has control over the FBI--oversight. So I have said, OK, let's keep the committee process in place, regular order, so if the FBI runs over to the Intel committee and they testify there, and they don't respond to my letter, I think that's a mistake, because the FBI is under our jurisdiction."
Graham said he'll give "them a little more time" but warned that (FBI Director James) Comey ought to respond to the letter.
"You'll run afoul of the Judiciary Committee. If I were you, I wouldn't do that," he said. "Director Comey, trying to give you a little advice here."
Trump has not publicly provided any evidence for his wiretapping
allegations, first made more than a week ago. Sunday night, Kellyanne Conway, a top adviser to the President, appeared to expand the allegations by suggesting to the Bergen Record
there could have been even wider spying of the Trump campaign, including the use of microwaves and television sets. Like the President, however, she did not provide any evidence, and later said she was speaking generally about surveillance.
Former President Barack Obama has denied the allegations through a spokesman, and his former director of national intelligence, James Clapper, also has publicly denied them. Sources have told CNN that Comey also pushed back against Trump's claim.
As Congress' deadline looms, calls for Trump to produce evidence have grown louder, including by some in his own party.
"The President has one of two choices: either retract or provide the information that the American people deserve," Sen. John McCain told CNN's Jake Tapper Sunday on "State of the Union." "I have no reason to believe that the charge is true, but I also believe that the President of the United States could clear this up in a minute."
Two congressmen privy to sensitive intelligence information said this weekend that they, too, had no information currently to support Trump's claims.
The House Intelligence Committee's top Democrat, Rep. Adam Schiff, said on ABC's "This Week": "I don't expect we'll see evidence of this."
And House Speaker Paul Ryan, asked directly on CBS News' "Face the Nation" if he had seen evidence of the wiretaps, said: "No."
Conway said Sunday that "surveillance" could go beyond the tapping of phones.
"What I can say is there are many ways to surveil each other," Conway told the Record. "You can surveil someone through their phones, certainly through their television sets — any number of ways."
She went on to allege that the spying could have been done with "microwaves that turn into cameras," adding: "We know this is a fact of modern life."
Conway also reiterated Trump's request that an investigation into the wiretapping claims be included into a congressional investigation into Russia's alleged meddling in the 2016 election.
Pressed on CNN's Chris Cuomo on "New Day" Monday morning about her comments to the Record, Conway said she was speaking about "surveilling generally."
"Chris, I'm not Inspector Gadget. I don't believe people are using the microwave to spy on the Trump campaign," Conway told CNN. "However, I have -- I'm not in the job of having evidence. That's what investigations are for. I've said many, many times throughout the week that the president is pleased that the House and Senate Intelligence Committees have agreed with him that this should be part of the investigation that already exists about Russia and the campaign, an investigation that apparently has gone nowhere so far."
She later tweeted
that the Record's headline -- "Kellyanne Conway suggests even wider surveillance of Trump campaign" -- was "just wrong."
"Response to Bergen Record was about surveillance articles in news & techniques generally, not about campaign. Headline just wrong," Conway said.
This story has been updated.