Trump left the meeting "leaning toward the former CT senator", according to the report.
Mr Lieberman was among four candidates Mr Trump interviewed at the White House on Wednesday.
"This is a moment where we need a law enforcement professional that's never campaigned for a presidential candidate, never campaigned for office, never worn a party label to head the FBI", McCaskill said. The attorney in question is former Democratic Senator Joe Lieberman.
Lieberman retired from the Senate in 2011, and since then has been a senior counsel at the NY law firm Kasowitz, Benson, Torres & Friedman LLP.
Trump and Sessions also interviewed former Oklahoma Gov.
Last week, the president justified his May 9 firing of FBI director James Comey based on a Justice Department memo accusing Comey of mishandling the 2015-16 investigation of former secretary of state Hillary Clinton's use of private email.
Now, the White House has pulled back that announcement, likely after seeing the incredible anger among Democrats who virulently oppose Lieberman - as they should. "Everybody likes and respects Joe Lieberman".More news: Biden: 2016 election tapped into "our darkest emotions"
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Legislators on both sides of the aisle have expressed wariness about having a politician lead the Federal Bureau of Investigation, and a Senate Democratic leadership aide said Lieberman would not be exempt from that.
Lieberman backed Sen. John McCain (R-Ariz.), a longtime friend, in the race for president in 2008.
When asked about possibility of him announcning his nominee by Friday, Trump said, "Even that is possible".
Also, some Democrats are skeptical about Lieberman's independence from Trump given that the two are now in an attorney-client relationship.
Mr Lieberman served in the Senate for more than two decades and was the Democratic vice presidential nominee in 2000 with Al Gore, then the sitting vice president. Tweeting on Friday afternoon, Sanders said that Lieberman failed to meet his bar because he would be "perceived as a political appointee". He served as a Democratic senator in CT for more than 20 years, from 1989 into 2013.
Many senators have said they will not support the nomination of a former politician. He also disputed the administration's characterisation of the investigation into potential coordination between Russian Federation and Trump associates.
Several other candidates have withdrawn from consideration: Rep. Trey Gowdy of SC and Sen.