The lawyer, Emmet T. Flood, met with Mr. Trump in the Oval Office this past week to discuss the possibility, according to the people. No final decision has been made, according to two of the people.

Should Mr. Flood come on board, the two people said, his main duties would be a day-to-day role helping the president navigate his dealings with the Justice Department.

Two people close to the president said that the overture to Mr. Flood did not indicate any new concerns about the inquiry. Still, it appears, at the least, to be an acknowledgment that the investigation is unlikely to end anytime soon.