While Cornyn is one of the more popular members of the Senate GOP Conference, a number of his colleagues are skeptical that he would be the right choice for the job, worried that such a pick would be viewed as too political at a time when many critics are questioning whether Comey was fired to slow-walk the FBI's investigation into alleged Russian coordination with the Trump campaign during last year's elections.
On Sunday, Senate Majority Leader Mitch McConnell declined to put his weight behind his top deputy, instead pushing another potential candidate, Merrick Garland
, who was former President Barack Obama's Supreme Court pick before the majority leader led an effort to scuttle his nomination last year. Sen. Lindsey Graham, who serves on the Senate judiciary committee with Cornyn, suggested that it made sense to name an FBI agent to the post -- rather than the Texas Republican.
And on Monday, Sen. Susan Collins, who sits on the intelligence committee with Cornyn, said it made sense to choose either Garland or Mike Rogers, a former House intelligence committee chairman and FBI special agent.
"I think the world of John Cornyn, and he would be a great choice in normal times, but we're not living in normal times," Sen. Susan Collins, a Maine Republican, told CNN Monday.
With Garland, Collins said: "No one could argue that he was beholden to anybody in any way and he would bring integrity and independence to the job."
On Monday, Cornyn, who has insisted he has not been offered the job, told CNN he interviewed this weekend for 45 minutes with Attorney General Jeff Sessions, one of his former Senate colleagues, and Rod Rosenstein
, who is the deputy attorney general. But Cornyn declined to say whether the Russia probe or Comey's firing were discussed. And he would not say if there was a pledge for loyalty, as Trump reportedly asked Comey
to commit to earlier this year, though the White House has denied that account.
"I met with the attorney and the deputy attorney general. I told him I was interested in having a conversation and, basically, that's what we did," Cornyn said.
Asked about the White House's shifting explanations for Comey's firing, Cornyn said he still backed the President's decision and supported the initial reasoning behind the firing -- that Comey mishandled the Clinton email investigation.
"I found the Rod Rosenstein email compelling -- that's still my view."
Cornyn would not comment about Trump's threats to release a tape of his conversations with Comey -- or if Congress should obtain those tapes. But Collins, like other Republicans, said the White House should be prepared to turn them over
"We have no idea whether or not there are tapes but if there are tapes then they should be turned over," Collins said. "That issue was settled many years ago when the Supreme Court ruled in 1974 that the tapes that President (Richard) Nixon had should be turned over to Congress."
Democrats have vowed to try to block the new FBI director
until Rosenstein names a special prosecutor to investigate the Russia probe. But they would need to court at least three Republicans to be successful. And Collins signaled she would not go that route.
"I don't think that's a good strategy at all," Collins said, adding that she wanted to question Rosenstein Thursday on whether a special prosecutor was indeed needed.