The former US senator from New Haven was one of four candidates Trump was interviewing Wedneday as possible replacements for ousted FBI Director James Comey.
Sarah Isgur Flores, a Justice Department spokeswoman, said Attorney General Jeff Sessions "has talked to all of" the remaining candidates about the position.
The naming of the special counsel has appeased some Democrats, who were threatening to try to block any Federal Bureau of Investigation director until such a move was made. He retired in 2013 as the US senator for CT after serving for 24 years.
"I hold Senator Lieberman's seat, he's no pushover, as both parties know".More news: Comey Hid Among White House Curtains to Avoid Trump
A November 25 news release from Trump's transition team quoted Lieberman praising Trump's selection of K.T. McFarland as deputy national security adviser.
Lieberman served four six-year Senate terms while living in New Haven's Westville neighborhood, three as a Democrat, then a final term as an independent after he lost the 2006 Democratic primary.
The White House has denied the content of the memo.
The president's nominee must be confirmed in the Senate, where Democrats and some Republicans have fiercely criticized Comey's firing. Lieberman gave reporters a thumbs-up as he left and said he and the president had a "good meeting".
Lieberman was also former Vice President Al Gore's running mate in the 2000 election.More news: Congressman Calls For Trump's Impeachment During House Floor Speech
The Senate must confirm Trump's candidate for the job.
Among their concerns was Lieberman's past praise of Michael Flynn, Trump's first national security adviser, who was sacked in February after misleading officials about his talks with Russian officials. He supported Sen. John McCain for president in 2008, but endorsed Hillary Clinton in 2016, a move that could count against him with Trump, who screens candidates for his administration for loyalty. John McCain and Lindsey Graham is the Democratic base's equivalent of a Tea Party candidate hosting a Planned Parenthood fundraiser.
He was the lead case agent on the Oklahoma City federal building bombing in 1995, when Keating was governor, and he was the FBI's on-scene commander after the September 11, 2001, attack on the Pentagon.
After leaving the office of governor, Keating served as the chairman of a panel appointed by the Catholic Church to investigate allegations of abuse by priests.
Keating told his hometown newspaper after the interview that he doesn't expect to be chosen.More news: Hunt for a place in finals begins as Mumbai play Pune
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