Speaking to reporters while meeting with Colombian President Juan Manuel Santos, Trump said he is "very close" to choosing a new FBI director to replace James Comey. And asked if Lieberman, the Democrat-turned-independent senator from Connecticut was a top candidate, Trump said yes.
Trump met with Lieberman Wednesday and found him "agreeable," a source said.
Trump and Attorney General Jeff Sessions have been interviewing candidates to lead the bureau since Trump suddenly fired former Comey last week only a few years into what was created to be up to a 10-year term.
The circumstances surrounding the firing have made the appointment closely watched, with senators calling for an unimpeachable, nonpartisan appointment.
The firing drew instant criticism from both sides of the aisle, as Comey was overseeing the investigation into Russia's meddling in the 2016 US election and any possible links between the Trump camp and Russia during the campaign.
While Trump's team initially claimed that the firing came after a review by Deputy Attorney General Rod Rosenstein, laid out in a memo, that largely faulted Comey on his handling of the investigation into the private email server of Hillary Clinton, Trump later said in an interview he had already decided to fire Comey and cited the Russia investigation as being at the top of mind.
After the firing, Rosenstein announced he would appoint a special counsel to oversee the investigation -- his decision because Sessions recused himself from it. Rosenstein only informed the White House of his decision to bring on former FBI Director Robert Mueller after the order was signed and less than an hour before it was announced publicly.
The naming of the special counsel has appeased some Democrats, who were threatening to try to block any FBI director until such a move was made. But the appointment does not assuage all concerns, and an FBI director would still play a major role as things moved forward.
Already one red-state Democrat, who would be a top target for Republicans to support a candidate, stated she would not be pleased with Lieberman as a pick.
"It's a mistake to nominate anyone who's ever run for office," Missouri Sen. Claire McCaskill told reporters. "I'm somebody who spent a lot of time in law enforcement; 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."
Lieberman's long history in politics
Lieberman was Democrat Al Gore's vice presidential nominee in 2000 and unsuccessfully ran for the Democratic presidential nomination in 2004.
But Lieberman, 75, is not necessarily popular with Democrats. He lost to a Senate primary challenger in 2006 and only retained his seat after he switched his party to independent. He also backed Republican Sen. John McCain's presidential bid in 2008 over Barack Obama, speaking at the Republican National Convention that year.
Since leaving the Senate, Lieberman has been a senior counsel at the New York law firm Kasowitz, Benson, Torres & Friedman LLP and serves on a number of boards.
Potentially complicating his issues with Democrats, Lieberman's firm has represented Trump in past litigation.
Other candidates considered included current acting Director Andrew McCabe, former Oklahoma Gov. and FBI Agent Frank Keating, career FBI official Richard McFeely and Texas Sen. John Cornyn and New York Judge Michael Garcia, who both withdrew their names.
When Cornyn was under consideration, even his GOP colleagues questioned whether he would be the right choice as an elected official.
Lieberman's closest law enforcement experience is serving as his state's attorney general, an elected position. FBI directors traditionally have had prosecutorial experience, either as an FBI agent or as a member of the Justice Department.
Lieberman was born in Stamford, Connecticut, educated at Yale and Yale Law school, and served in the Connecticut state senate and as attorney general of the state before being elected as the state's Senator in Washington.
First elected in 1988, Lieberman served as a Democrat until 2006, when he became an independent who caucused with the Democrats. He did not seek re-election in 2012.
Lieberman is perhaps most famous for joining the ticket with Gore in his unsuccessful run for president in the 2000 election. Lieberman's selection as a vice presidential candidate was the first time a Jewish person was on a major party's presidential ticket.
The former senator was known to buck his party even before endorsing McCain. Lieberman was the first Democratic senator to criticize former President Bill Clinton publicly during the Monica Lewinsky scandal.
Lieberman is a devout follower of Orthodox Judaism.
Clarification: This story has been updated to clarify the question Trump was asked about Lieberman.