According to sources around Comey, he refused the request, although Trump denies having asked.
He spoke following Mr Trump's sudden firing of Mr Comey last week, which drew sharp criticism because it came amid the FBI's investigation into Russia's interference in the 2016 presidential election.
Mr Comey promised only that he could be honest, The New York Times reported.
It comes as the search intensifies to replace fired FBI Chief James Comey.
Democrats would need Republican support to follow through because it takes 51 votes for confirmation - and Republican now hold 52 seats.
What exactly would Comey tell us and the Sentate Intelligence Committee? "We're hoping. We're waiting".
Comey's associates sayhe is not seeking publicity and that they believe that an open session before Congress is the most appropriate setting.
USA intelligence agencies have made clear their conclusion that Russian Federation meddled in the 2016 election.
CLAPPER: The founding fathers, in their genius, created three co-equal branches of government, and a built in system of checks and balances, and I feel as though that is under assault, and is eroding. "I think it's a good idea", he told Pirro on her show "Justice with Judge Jeanine" that aired Saturday.
After comments that the administration intends to move "very quickly" on the process, a reporter in the White House press pool asked the president if that could mean finding a permanent replacement to spearhead the agency by the end of the week. Since then, a lot of controversy has surrounded the president's decision.
The South Carolina Republican sits on the Senate Judiciary Committee.
Rogers is the former chairman of the House Intelligence Committee. John Cornyn of Texas.
They are among a dozen candidates being considered for the job.
Comey is said to be concerned with the prospects of numerous initiatives he was pursuing inside the bureau, but believes FBI agents and personnel will continue to conduct themselves with professionalism. The FBI is investigating the Russians' involvement.
Critics have alleged that President Donald Trump's firing of FBI Director James Comey was an effort to stifle the FBI probe. They are among almost a dozen candidates Trump is considering, a group that includes several lawmakers, attorneys and law enforcement officials.
The White House said in a statement that Comey was terminated and removed from office based on "the clear recommendations of both Deputy Attorney General Rod Rosenstein and Attorney General Jeff Sessions".
Those interviewed for the job include Sen. Acting FBI Director Andrew McCabe was also interviewed, despite his repeated willingness to break from the White House explanations of Comey's ouster.
The others are Alice Fisher, a white-collar defense lawyer who previously led the Justice Department's criminal division, and Michael J. Garcia, a judge on the New York State Court of Appeals who previously served as the us attorney for the Southern District of New York.
Schumer, while proffering no names, agreed that the next Federal Bureau of Investigation leader should be someone of strong character but no strong political leanings.
HTC unveils new flagship U11 smartphone
The device will come with inbuilt Google Assistant in some regions and in some other markets it will also feature Amazon Alexa. HTC is said to ditch the 3.5mm headset jack in U 11, which will come with USB Type-C for charging as well as audio output.
Kimmel to return as Oscar host next year
Jimmy Kimmel as the Oscars host in 2017 didn't help the annual awards ceremony to reverse it's continuing downward ratings trend. The 90th Oscars will take place on March 4, 2018 in Los Angeles and will be broadcast live on ABC.