In a series of tweets, White House press secretary Sean Spicer called Mr Trump's allegation, for which he has provided no evidence, "very troubling".
He said Mr Trump is requesting Congress examine whether "executive investigative powers were abused" during the 2016 election campaign, as part of an ongoing congressional probe into Russia's influence on the election.
"Reports concerning potentially politically motivated investigations immediately ahead of the 2016 election are very troubling," Mr Spicer tweeted.
"President 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."
He added: "Neither the White House nor the President will comment further until such oversight is conducted."
Over the weekend, Mr Trump called Mr Obama a "bad, or sick, guy" after accusing him of ordered his offices in New York be wiretapped during the US election.
"Terrible!" the President tweeted over the weekend. "Just found out that Obama had my 'wires tapped' in Trump Tower just before the victory. Nothing found. This is McCarthyism!"
Soon after he added: "Is it legal for a sitting President to be 'wire tapping' a race for president prior to an election? Turned down by court earlier. A new low!
"I'd bet a good lawyer could make a great case out of the fact that President Obama was tapping my phones in October, just prior to election!"
Despite the severity of his claim, Mr Trump did not provide any evidence Mr Obama was responsible for surveillance at his property and has not provided any since.
White House officials have reportedly said they have no idea where Mr Trump got his wiretapping claims from.
Two former senior US officials dismissed Mr Trump's accusations out of hand as “just nonsense” and “just wrong”, with one telling CNN categorically: “This did not happen”.
One White House official is reported to have “grimaced” when he woke up and saw the President’s flurry of tweets, according to Politico.
“It could have come from anywhere”, the official reportedly said, adding it was unlikely to have been an official source.
Mr Obama's spokesman denied the allegations, saying any suggestion he or his staff "ordered surveillance on any US citizen" was false.
Kevin Lewis added that a “cardinal rule” of the Obama administration was that no White House official ever interfered in any Justice Department investigations, which are supposed to be conducted free of political influence.
“As part of that practice, neither Obama nor any White House official ever ordered surveillance on any US citizen. Any suggestion otherwise is false,” he said.
The statement did not address the possibility that a wiretap of the Trump campaign could have been ordered by Justice Department officials.
Under US law, a federal court would have to have found probable cause that the target of the surveillance was an “agent of a foreign power” in order to approve a warrant authorising electronic surveillance.
James Clapper, the former director of national intelligence in the Obama administration, said: "there was no such wiretap activity mounted against the president, the president-elect at the time, as a candidate or against his campaign."
He said as intelligence director he would have known about such an order, and added: "Absolutely, I can deny it."
White House spokeswoman Sarah Huckabee Sanders went on ABC's This Week to call for a congressional investigation into allegations the Obama administration ordered the wiretaps.
However, Ms Sanders refused to say where Mr Trump got his information from or why he blamed the former president.
She told this week: "If they're going to investigate Russia ties, let's include this as part of it. That's what we're asking."
Ms Sanders would not elaborate on what the President meant, saying his tweets speak for themselves. She also would not say exactly where the President got his information.
Without being specific, Ms Sanders said Mr Trump is "going off information that he's seen that has led him to believe that ... And if it is, this is the greatest overreach and the greatest abuse of power that I think we've ever seen and a huge attack on democracy itself."
The top House Democrat said it is "just ridiculous" for Mr Trump to claim Mr Obama would ever have ordered any wiretap of an American citizen.
Representative Nancy Pelosi said "we don't do that" and she called the charge a "smear."
Ms Pelosi told CNN's State of the Union that Mr Trump is following the playbook of making something up, having the media report it and then saying everybody is writing about it.
The California Democrat said it was a "tool of an authoritarian" to always have people "talking about what you want them to be talking about."
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