Republican lawmakers said Sunday that they will include President Trump’s explosive claim that the Obama administration engaged in politically driven wiretapping in the congressional probes of Russian campaign meddling.

Rep. Devin Nunes, the California Republican who heads the House Permanent Select Committee on Intelligence, said the president’s allegation would be examined as part of the panel’s investigation.

“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,” Mr. Nunes said. “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.”

He made the statement after the White House issued a request for Congress to investigate whether the Obama administration was involved in wiretapping Trump Tower during the election.

“Reports concerning potentially politically motivated investigations immediately ahead of the 2016 election are very troubling,” the White House said in a brief statement.

“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,” the statement said.

The White House submitted its request a day after Mr. Trump stunned Republicans and outraged Democrats by saying in a tweet, “Terrible! Just found out that Obama had my ‘wires tapped’ in Trump Tower just before the victory. Nothing found. This is McCarthyism!”

The White House statement concluded, “Neither the White House nor the president will comment further until such oversight is conducted.”

Mr. Trump’s wiretapping claim outraged Democrats, who demanded evidence to back up the charge.

“I can tell you that it’s just ridiculous for President Trump to say that President Obama would ever order any wiretap of an American citizen, any president,” House Minority Leader Nancy Pelosi, California Democrat, said on CNN’s “State of the Union.” “That’s just — we don’t do that.”

Senate Minority Leader Charles E. Schumer, New York Democrat, said any wiretapping would be a problem for Mr. Trump, not Mr. Obama, because it would show that a federal judge found probable cause to allow the surveillance.

“Either way, Chuck, the president’s in trouble,” Mr. Schumer told NBC’s Chuck Todd on “Meet the Press.”

By refusing to comment pending a congressional investigation, the Trump administration took itself out of the direct line of political fire while guaranteeing that speculation will continue to brew over whether the Democratic administration crossed the line into campaign dirty tricks.

Moving to flatten such conjecture were top officials from the Obama administration, led by former Director of National Intelligence James R. Clapper, who insisted Sunday that the Obama administration did not seek or receive a special warrant to monitor electronically communications from Trump Tower after Mr. Trump’s shocking Nov. 8 victory.

“I can deny it,” Mr. Clapper said on NBC’s “Meet the Press.”

He referred to claims that the Obama White House sought or obtained a surveillance warrant from the secretive court established under the Foreign Intelligence Surveillance Act, or FISA, to track suspected communication between Trump officials and Russia.

Domestic surveillance, including wiretapping, by American intelligence can be legally conducted only after obtaining a warrant from the classified FISA court.

“There was no such wiretap activity mounted against the president-elect at the time as a candidate or against his campaign,” Mr. Clapper said.

The FBI also reportedly got involved during the weekend. The New York Times and CNN reported Sunday that the bureau asked the Justice Department to rebut publicly Mr. Trump’s claims of wiretapping.

Citing “two sources with knowledge of the situation,” CNN reported that the FBI wants a public disavowal of what would be an illegal act if it had happened.

Neither the Justice Department nor the FBI had any public comment Sunday.

Kevin Lewis, a spokesman for Mr. Obama, denied that his White House had ordered surveillance on Mr. Trump.

“A cardinal rule of the Obama administration was that no White House official ever interfered with any independent investigation led by the Department of Justice,” Mr. Lewis said. “As part of that practice, neither President Obama or any White House official ever ordered surveillance on any U.S. citizen. Any suggestion otherwise is simply false.”

But former Obama speechwriter Jon Favreau threw in a caveat, saying Saturday on Twitter, “I’d be careful about reporting Obama said there was no wiretapping. Statement just said that neither he nor the WH ordered it.”

Republicans steered clear of the merits of Mr. Trump’s accusation, saying they would wait for results of the congressional investigations.

Sen. Marco Rubio, a Florida Republican who sits on the Senate Select Committee on Intelligence, said he had no details about Mr. Trump’s claim but “perhaps the president has information that is not yet available to us or to the public.”

“What I think we should do is, everybody needs to take a deep breath and calm down here,” Mr. Rubio said on CNN’s “State of the Union.” “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.”

He said the committee would issue a report, “and then people can form judgments on the basis of the collection of facts.”

Sen. Tom Cotton, Arkansas Republican, said the Senate probe would include “any allegations such as this.”

Mr. Cotton said he had seen no evidence of such claims, “whether that’s a potential FISA court application or denial of that application, or a resubmission of an application or surveillance, for that matter.”

“That doesn’t mean that none of these things happened,” Mr. Cotton told “Fox News Sunday.” “It simply means I haven’t seen that yet.”

Mr. Trump posted the series of tweets after conservative radio talk show host Mark Levin made the case on his Thursday night show that the Obama administration requested FISA warrants on two occasions last year to monitor Trump Tower over concerns about links to Russian banks.

He cited articles in Heat Street, which reported that sources in the counterintelligence community said the administration’s FISA application was rejected in June but that a more narrowly tailored request was accepted in October.

“The evidence is overwhelming. This is not about President Trump’s tweeting. This is about the Obama administration’s spying,” Mr. Levin said Sunday on “Fox & Friends.”

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