In the letter, Lieberman cites the decision by the White House to enlist Marc Kasowitz as outside counsel to coordinate the administration's response to the Russia investigation, saying, "With your selection of Marc Kasowitz to represent you in the various investigations that have begun, I do believe it would be best to avoid any appearance of a conflict of interest, given my role as a senior counsel in the law firm of which Marc is the senior partner."
Last week Trump said he was "very close" to choosing a new FBI director.
According to sources, some on Trump's team thought Lieberman would be a good choice and would appeal to Democrats.
Others pointed out that some Democrats dislike Lieberman (in 2006 he lost a Democratic Senate primary race, ran as an independent in the general contest and won re-election, and in 2008 he endorsed his friend Sen. John McCain, R-Ariz., for president) and that the wiser move would be to appoint a career FBI person or somebody else with a law enforcement background — a choice with credibility in the bureau and on Capitol Hill.
Reactions to reports that Lieberman was in the running were mixed. Sens. Brian Schatz, D-Hawaii, and Claire McCaskill, D-Mo., called on Trump to choose an FBI director with experience in law enforcement.
"It is a very bad idea to appoint a politician to head the FBI right now. We need a law enforcement professional. #bipartisan support," McCaskill wrote on Twitter.
McCain took offense at the suggestion.
"Joe Lieberman has more experience than all of my Democratic colleagues combined. So screw them. And you can quote me," he said on May 18.
The search for a new FBI director comes after Trump's abrupt firing of Comey this month amid an investigation into possible Trump campaign associates' ties to Russian officials.