Previous media reports have suggested that the Foreign Intelligence Surveillance Act (Fisa) court granted a warrant to enable the FBI to conduct surveillance of "US persons" in an investigation of possible contacts between Russian banks and the Trump Organization.
Conway on Monday cited "credible news sources".
"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", Obama spokesman Kevin Lewis said in a statement Saturday, Yahoo! reports.
Which is true - at least when it comes to top-level sensitive information gathered by US intelligence agencies. Columnist Isabella Winkler claims that Trump fighting back through his tweets is an unneeded and unpresidential move.
On Monday, Mr Schumer also called for the DoJ's independent inspector-general to investigate whether the White House has interfered with the investigation of the Trump campaign's ties to Russian Federation.
"The hypocrisy on this story is outrageous - we've dealt with this for the last six months", she said. A US Justice Department spokesperson also declined to comment. However, news of FISA orders and transcripts of telephone conversations and wiretaps are "almost like an alternative government working against the president".
Obama has already refuted Trump's claim as "simply false".
The White House on Sunday asked the U.S. Congress to examine whether the Obama administration abused its investigative authority during the 2016 campaign, as part of the ongoing congressional probe into Russia's influence on the election. James Clapper, former director of national intelligence, said Sunday that "there was no such wiretap activity mounted against the president-elect, at the time, as a candidate or against his campaign".
Trump is thought to have accused his staff of fumbling the situations with Sessions and took out his fury the day after his recusal.
But some Republican lawmakers appear to be losing patience with the drama of Trump's presidency.
"I'm nearly 100 percent certain" that Trump and Comey have not spoken since Saturday, White House press secretary Sean Spicer told reporters Monday in an off-camera briefing. This is not a small detail, since any White House knowledge about, to say nothing of direction of, this type of surveillance would be illegal.
"This is called the wrap-up smear", she told CNN. Marco Rubio said on "Meet The Press" on Sunday. "It's a tool of an authoritarian to just have you always be talking about what you want them to be talking about".
The White House did not immediately respond to requests for comment. It will all come out. I haven't seen him this p****d off in a long time.
"An "extremely credible source" has called my office and told me that @BarackObama's birth certificate is a fraud", Trump tweeted in August 2012, at the height of the presidential race between Obama and Republican Mitt Romney. The White House has denied any such links with Ms Sanders saying the press is peddling a "false narrative that there are connections between the Trump campaign and the Russians".