Senate Judiciary Committee chairman Charles Grassley is holding up the confirmation of deputy attorney general nominee Rod Rosenstein until Mr Comey testifies before his committee on the scope of the FBI's Russian Federation probe.
But the press secretary refused to acknowledge Nunes, who was on Trump's transition team, had also said there was no proof of wiretapping.
Burr and Warner were among eight senior congressional leaders briefed Friday by FBI Director James Comey.
While he had offered no evidence, the president can access information on law enforcement and intelligence investigations, and possibly possessed some classified information supporting the claim.
A bipartisan group of lawmakers on the House and Senate intelligence committees said this week they have not seen anything to support his allegations.
But White House spokesman Sean Spicer said on Thursday the United States president maintained the allegation, adding: "He stands by it". GOP Chairman Rep. Devin Nunes said he did not "think there was an actual tap of Trump Tower."
"Because we need at least those in the global community to be able to take the words of the president seriously, and unfortunately, we've learned today on a bipartisan basis that that can not be done".
House Minority Leader Nancy Pelosi said Mr. Trump's claims were "irresponsible for sure".More news: Paul Ryan's health-care ordeal probably won't end well
Obama flatly rejected it, but the White House refusal to back down has kept the issue alive.
Spicer also claimed earlier this week that Trump had accused Obama of generally surveilling him, rather than of specifically wiretapping Trump's phones. Today, Conway backed off the claim telling CNN's Chris Cuomo, "I don't believe people are using the microwave to spy on the Trump campaign".
"We said nothing", Trump noted. "All we did was quote a certain very talented legal mind who was the one responsible for saying that on television".
Trump tweeted earlier this month that his Democratic predecessor had wiretapped him during the late stages of the 2016 campaign. That really covers, because wiretapping is pretty old fashioned stuff.
However he did not provide any evidence to support his claim, and thus far no evidence is thought to have been produced. "I think you're going to find some very interesting items coming to the forefront over the next two weeks", he said.
'And I will be perhaps speaking about this next week'.
Senate Intelligence Committee Chairman Richard Burr, a Republican from North Carolina, and Vice Chairman Mark Warner, a Democrat from Virginia, jointly said there were "no indications" that Trump Tower, the Manhattan building where Mr. Trump lived and worked prior to winning the presidency, was under any form of government surveillance. In a statement to The Associated Press, Stone said he's retained two attorneys to explore whether he can compel the government to "either charge me or admit they have no case whatsoever".