The request came a day after President Donald Trump alleged, without supporting evidence, that then-President Obama ordered a wiretap of the phones at Trump's campaign headquarters in Trump Tower in New York.
Mr Trump, who has been facing intense scrutiny over alleged Russian interference in support of his presidential bid, made the wire-tapping allegation in tweets written from his weekend home in Florida early on Saturday.
His press secretary said the inquiry into alleged Russian interference should also examine these allegations.
James Clapper, the US intelligence chief during the campaign, has flatly denied Trump Tower was wire-tapped. Mr Obama's spokesman, Kevin Lewis, had earlier said that the former president had never "ordered surveillance of any US citizen".
Mr Trump called the alleged tapping "Nixon/Watergate", referring to the notorious political scandal of 1972, which led to the downfall of President Richard Nixon.
His claims sparked Republican and Democrat politicians alike to demand details to back them up. Florida Republican Senator Marco Rubio was the latest, saying on Sunday that "the White House will have to answer as to exactly what he was referring to".
But in a series of tweets on Sunday, White House Press Secretary Sean Spicer did not provide any further evidence.
He said: "Reports concerning potentially politically motivated investigations immediately ahead of the 2016 election are very troubling.
"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.
"Neither the White House nor the President will comment further until such oversight is conducted."
Wire-tap claim denied by ex-intelligence chief
James Clapper, who left his post when Mr Trump took office on 20 January, told NBC's Meet the Press: "There was no such wire-tap activity mounted against the president-elect at the time, as a candidate, or against his campaign."
He said that as intelligence director he would have known about any "court order on something like this. Absolutely, I can deny it."
Some media reports had suggested the FBI had sought a warrant from the foreign intelligence surveillance court (Fisa) in order to monitor members of the Trump team suspected of irregular contacts with Russian officials.
Mr Clapper's comments appear to contradict the reports, which said that a warrant was at first turned down, but then approved in October.
Under Fisa, wire-tapping can only be approved if there is probable cause to believe that the target of the surveillance is an agent of a foreign power. Mr Obama could not lawfully have ordered such a warrant.
Mr Trump's tweets followed allegations made by conservative radio host Mark Levin, including that the Obama administration "sought, and eventually obtained, authorisation to eavesdrop" on the Trump campaign last year.
Mr Obama's spokesman, Kevin Lewis, said the accusation was "simply false".
Mr Lewis said the accusation was "simply false".
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", he said.
The statement left open the possibility that a judicial investigation had been taking place.
Some Democrats have suggested Mr Trump's allegations were an attempt to focus attention away from the Russian affair.
House Democratic Leader Nancy Pelosi said: "The Deflector-in-Chief is at it again."
Photo: AFP (Donald Trump, then president-elect, and Barack Obama in the Oval Office at the White House in November 2016).