Mr. Kushner’s comments underscored a key factor that was lacking in Mr. Trump’s relationships with his first two chiefs of staff, Reince Priebus, the former Republican National Committee chairman, and Mr. Kelly: personal chemistry.

The president posted on Twitter on Tuesday morning that “many, over ten, are vying for and wanting the White House Chief of Staff position,” pushing back against reports that he is having trouble recruiting.

In reality, the number of people actively expressing interest is far smaller.

David Bossie, a former campaign aide, is publicly campaigning for the job, and Representative Mark Meadows, a hard-right Republican congressman from North Carolina, has made it clear that he would be interested in the position if it were offered to him. But others who have been considered — like Steven Mnuchin, the Treasury secretary; Robert Lighthizer, the United States trade representative; and Mick Mulvaney, the director of the Office of Management and Budget — have so far made it clear that the timing is not right for them to leave their current posts.

Mr. Bossie, who is scheduled to meet on Friday with Mr. Trump at the White House, appeared to be making a pitch based on his experience with the congressional oversight process, gleaned from his work as a House investigator during President Bill Clinton’s impeachment.

In an interview with Mr. Trump’s favorite morning program, “Fox & Friends,” Mr. Bossie said the president needed a chief who would be “able to handle what is the subpoena cannon — somebody who will understand what the House is going to do to him and White House staff in trying to drag them into a legal process and an investigative process that will slow down the success of this White House.”

For his part, Mr. Meadows described himself as “more of a strategy guy” to reporters on Monday.

People close to Mr. Trump said that he will almost certainly want assurance that someone would accept the job before he actually offers it.