Republicans, meanwhile, urged Trump to let the House and Senate intelligence committees do their jobs.
Both parties made clear that Trump's allegation -- which the White House has not offered any evidence to support, and which Obama administration officials emphatically deny -- won't be dropped anytime soon, as even GOP lawmakers who have backed Trump said they'd investigate.
Rep. Devin Nunes of California, the Republican chairman of the House Intelligence Committee, said his committee would look into whether the federal government was conducting wiretapping of campaign officials as part of the panel's investigation into Russian meddling in the US election.
"One of the focus points of the House Intelligence Committee's investigation is the U.S. government's response to actions taken by Russian intelligence agents during the presidential campaign," Nunes said in a statement Sunday. "As such, the Committee will make inquiries into whether the government was conducting surveillance activities on any political party's campaign officials or surrogates, and we will continue to investigate this issue if the evidence warrants it."
Democrats on the House Judiciary Committee announced Sunday night they planned to ask White House Counsel Donald McGahn to provide the committee with a list of communications between the White House and the Department of Justice regarding probes into Russian interference in the election.
Other Democrats, meanwhile, accused Trump of advancing yet another unfounded conspiracy in an effort to distract from the issues dogging his administration in its early weeks.
"You make up something and then you have the press write about it, and then you say, 'everybody's writing about this charge,'" House Democratic leader Nancy Pelosi said on CNN's "State of the Union." "It's a tool of an authoritarian, to just have you always be talking about what you want to be talking about."
Their comments come after Trump's aides asked Congress to look into whether the Obama administration abused its investigative powers during the 2016 election.
"Reports concerning potentially politically motivated investigations immediately ahead of the 2016 election are very troubling," White House press secretary Sean Spicer said in the statement, which he also posted on Twitter. "President Donald J. Trump is requesting that as part of their investigation into Russian activity, the congressional intelligence committees exercise their oversight authority to determine whether executive branch investigative powers were abused in 2016.
"Neither the White House nor the President will comment further until such oversight is conducted," Spicer added. He did not provide any further details on the President's request to Congress.
While Spicer said "reports" prompted the call for a congressional investigation, the White House still has not provided any evidence to back up the President's accusations. There are no publicly known credible reports to back up Trump's claim that Obama ordered Trump's phones be monitored.
Frustrated that the Russia stories have overshadowed a widely praised performance in his joint address to Congress on Tuesday, Trump angrily raised the wiretapping issue unprompted in conversations with friends and acquaintances at Mar-a-Lago, two people who have spoken with him at his Palm Beach resort over the last 24 hours said on Sunday. The President didn't specify what information he was basing his accusations upon, but told them he expected an investigation to prove him right.
Newsmax CEO Christopher Ruddy posted a column Sunday saying he spoke with the President twice on Saturday about the wiretap story.
"When I mentioned Obama 'denials' about the wiretaps, he shot back: 'This will be investigated, it will all come out. I will be proven right," Ruddy wrote.
Multiple former senior US officials have dismissed Trump's allegations, however, calling them "nonsense" and "false." Obama, through a spokesman, also rejected the claim that he ordered Trump's phones tapped.
Former Director of National Intelligence James Clapper, a career intelligence official who had oversight of the US intelligence community in that role, said Sunday that Trump was not wiretapped by intelligence agencies nor did the FBI obtain a court order through the Foreign Intelligence Surveillance Act to monitor Trump's phones.
"For the part of the national security apparatus that I oversaw as DNI, there was no wiretap activity mounted against the President-elect at the time, or as a candidate, or against his campaign," Clapper said Sunday morning on NBC's "Meet the Press."
Members of the intelligence committees
House Intelligence Committee ranking member Adam Schiff called Trump's claims as "destructive as it was baseless," but tied the President's call to the California Democrat's criticism of FBI Director James Comey for not being forthcoming about any investigation of ties between Russia and the Trump administration.
"If the administration truly believes that President Obama illegally eavesdropped on the Trump campaign and wants our committee to investigate the matter, they should join my call on Director Comey to answer any question put to him that is pertinent to the Russia investigation," Schiff said in a statement.
The chairman of the Senate Intelligence Committee, Republican Richard Burr of North Carolina, said his panel would follow the evidence trail in its investigation of Russian intelligence activities.
"As I've said since the beginning and have repeated since, the Senate Select Committee on Intelligence will follow the evidence where it leads, and we will continue to be guided by the intelligence and facts as we compile our findings," Burr said.
Rep. Jim Himes, a Democrat from Connecticut who sits on the House Intelligence Committee, tweeted at Spicer that he looks forward "to seeing your evidence," calling the allegations a "mess."
Himes also mocked Trump's claim in a tweet Sunday.
"BREAKING: HIMES ADMITS TO TAPPING. 'I know this will end up being sticky, but yes, I did the tapping' #maplenotwire" he tweeted
with a picture of him tapping a tree for syrup.
Sen. Marco Rubio, a Republican who sits on the Senate Intelligence Committee, said Sunday on CNN's "State of the Union"
that he's "not sure what it is (Trump) is talking about," but added it's possible that "perhaps the President has information that is not yet available to us or to the public."
Rubio pointed out that the Senate Intelligence Committee is already in the process of reviewing Russia's meddling in the US elections last year and will eventually issue a public report.
"But what I think we should do is, everybody needs to take a deep breath and calm down here," the Florida Republican told CNN's Jake Tapper. "And let's go through this as what we are doing. In the Senate Intelligence Committee, we are working in a bipartisan way to collect facts that involve reviewing classified and sometimes unclassified and open-source information."
Sen. Susan Collins, also a member of the Senate Intelligence Committee, said Sunday morning that she has "seen no evidence" to back up Trump's wiretapping allegation and called on the White House to turn over any evidence to the committee.
"What we need to deal with is evidence, not statements," the Maine Republican said on CBS' "Face the Nation."
Senate Minority Leader Chuck Schumer said he believes Trump is "in trouble" whether there was a court order authorizing surveillance of his phones or "if he falsely spread this kind of misinformation."
"If he falsely spread this kind of misinformation, that is so wrong. It's beneath the dignity of the presidency. It is something that really hurts people's view of government," the New York Democrat said Sunday on "Meet the Press." "On the other hand, if it's true, it's even worse for the President. Because that means that a federal judge, independently elected, has found probable cause that the president, or people on his staff, have probable cause to have broken the law or to have interacted with a foreign agent. Now that's serious stuff."
Schumer said he had no problem with the congressional intelligence committees including Trump's request in their investigation into Russian activity in the 2016 election, but he also used the call as an opportunity to reiterate Democrats' calls for a special prosecutor to investigate connections between the Trump campaign and Russian officials.