"The FBI Director and others have made it very clear that the president is not the subject of anything".
Warner also said he hopes to have Comey testify in a public hearing before his committee.
"The key is getting some of our Republican colleagues to join us", Schumer said.
Less than a week after Trump fired Comey, the administration has interviewed at least eight candidates to be Federal Bureau of Investigation director, and Trump has said a decision could come before he leaves Friday on his first overseas trip as president.
Comey was overseeing two important investigations that shaped Republican Trump's past year victory against his Democratic rival Hillary Clinton. Lee said he was "absolutely serious about it".
Sessions has faced questions over whether his involvement in Comey's firing violates his pledge to recuse himself from investigations into Russian interference in the election. Eight met at the Justice Department on Saturday with Attorney General Jeff Sessions and his deputy, Rod Rosenstein.
Thirty-four percent said Trump was concerned with Comey's investigation of his administration's ties to Russian Federation.
Two new names, however, emerged on Saturday: FBI special agent Adam Lee and U.S. District Judge Henry Hudson.
"There are a lot of Democrats who feel that way", Schumer said.
"I believe that we should not actually vote on an FBI Director replacement until there is a special prosecutor", Gillibrand said.
Asked whether Trump had fired Comey in order to obstruct ongoing investigations, Warner said "what was going on the president's mind, I don't know". "We'll have to discuss it as a caucus, but I would support that move". Trump later tweeted that Comey should be careful about "leaking to the press", in case "tapes" of conversations between Comey and Trump exist.
Republican Senator Lindsey Graham of SC said the White House must "clear the air" about whether there are any taped conversations.
Cornyn is the second-ranking Senate Republican and a strong defender of Trump, a president who often places a premium on loyalty.
Initial contenders for the job, according to a person familiar with the matter, included Cornyn, acting FBI Director Andrew McCabe, former Justice Department official Alice Fisher, and Michael Garcia, a former U.S. Attorney in Manhattan. The tweet was widely interpreted as a threat to Comey. Sen.
Senate Democratic Leader Chuck Schumer said his caucus has not yet made a decision on whether to withhold their votes, but added that the issue is being looked at as a way to ensure there is a thorough investigation of alleged Russian meddling in the 2016 US election.
"We have got to make sure that these tapes, if they exist, don't mysteriously disappear", he said.
If there are recordings, Republican Senator Mike Lee of Utah told the "Fox News Sunday" program it was "inevitable" that they would be subpoenaed and the White House would have to release them.
"If there are any tapes of this conversation, they need to be turned over", Senator Lindsey O. Graham, R-S.C., said on NBC's "Meet the Press". "You can't be cute about tapes". That may help explain why the scandal has not affected Trump's (already low) approval ratings, which are now at 39 percent, one point lower than what was recorded in last month's poll.