Last Updated May 18, 2017 7:14 PM EDT
Former vice presidential candidate and Connecticut Senatoras FBI director, but there is some hesitation because of his maverick approach to politics.
President Trump and his inner circle don't know how to judge Lieberman's past as a Democrat and independent. This hasn't disqualified him, but it remains a barrier Lieberman has to surmount -- if he can or if he wants to. Mr. Trump and Attorney General Jeff Sessions met with Lieberman at the White House on Wednesday. Sen. John McCain, in an appearance on MSNBC's "Morning Joe," gave Lieberman an enthusiastic endorsement, saying that he "is not only respected and admired, but beloved by all of us on both sides of the aisle." McCain, often a vocal critic of the president, suggested he may not have helped Lieberman's cause.
Mr. Trump, asked on Thursday if Lieberman is a top contender, said, "He is."
But Democrats are opposed to Lieberman, seeing him as too political. None of the Democrats CBS News spoke with Thursday backed him for FBI director.
A senior senate democratic aide told CBS News "none" of the Democrats in Congress support Lieberman.
"There is overwhelming opposition to Senator Lieberman in the Democratic caucus, and nobody thinks that he is going to be a fair and down-the-middle FBI director," the aide said.
"This is exactly the wrong time to take the unprecedented step to put a politician in charge of the FBI," the aide added.
Missouri Sen. Claire McCaskill said she thought it "would be a mistake to nominate anyone who's ever run for office."
Lieberman served as Connecticut's attorney general prior to winning his senate seat in 1988, but he has no previous FBI experience. He was the Democratic vice presidential candidate in 2000, but lost his primary in 2006 and returned to the Senate as an independent. He retired from office in 2012. During the 2016 presidential election, he endorsed former Democratic presidential candidate Hillary Clinton.
A number of members of Congress and other Republican and Democratic leaders have also called the White House to voice their support of former Michigan congressman and FBI agent Mike Rogers for director. The FBI rank and file also like the idea of Rogers as the next FBI chief, but several lawmakers have been told by the White House that "it helps us to have him out there, but he is not our kind of FBI director."
Mr. Trump and Sessions also-- former Republican Oklahoma governor Frank Keating, acting FBI Director Andrew McCabe, and former FBI official Richard McFeely.
CBS News Washington bureau chief Chris Isham, CBS News legal correspondent Paula Reid and CBS News congressional correspondent Nancy Cordes contributed to this report.