President Trump, 24 hours from his self-imposed deadline for picking a new F.B.I. director, told reporters on Thursday that he was “very close” to choosing a successor to James B. Comey, and he named Joseph I. Lieberman, the former Democratic senator and vice-presidential nominee, as a finalist.

But members of Mr. Trump’s staff — alarmed by his rapid embrace of Mr. Lieberman, a charming 75-year-old political operator with no federal law enforcement experience — have quietly urged him to take more time to make such a critical hire. By late Thursday, the president appeared increasingly likely to leave Friday for a nine-day foreign trip without picking a new director, according to three senior administration officials speaking on the condition of anonymity.

Mr. Lieberman, who served three terms in the Senate as a Democrat and one as an independent, would be an atypical choice to lead the F.B.I., whose agents prize the bureau’s independence as one of Washington’s few apolitical institutions. Judges and former prosecutors, not elected officials, have frequently been chosen.

Administration officials described the search as fluid and said the president and his team were keeping the decision-making process closely held to avoid the leaks that Mr. Trump believes are endemic to the West Wing.