Former Senator Joe Lieberman is apparently no longer under consideration for FBI director, a source familiar with the matter confirms to CBS News' Pat Milton.

The source said that the White House is continuing its search for an FBI director to replace James Comey who was fired by President Trump earlier this month.

Lieberman, who represented Connecticut in the Senate as a Democrat and is now an independent, was believed to be the frontrunner for the nation's top law enforcement job.

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One potential complication -- the Washington Post reported Tuesday that Mr. Trump hired Marc Kasowitz as his private attorney to deal with the federal investigation into Russia's interference in the 2016 U.S. election, and Lieberman has been working as a senior counsel at Kasowitz's law firm for several years.

Mr. Trump had suggested that he could announce his nominee before he left for his first overseas trip as president last Friday, but the announcement never came.

The president had also been considering Republican Oklahoma governor Frank Keating, acting FBI Director Andrew McCabe and former FBI official Richard McFeely. Other candidates have included former DOJ official Alice Fisher; Adam Lee, special agent in charge of FBI Richmond; New York State Judge Michael Garcia, Federal Judge Henry Hudson, former Homeland Security Adviser Fran Townsend and Mike Rogers, former chairman of the House Intelligence Committee.

Senate Majority Whip John Cornyn, R-Texas, withdrew himself from consideration last week.

CBS News' Pat Milton contributed to this report.