Mr. Christie has been a friend of Mr. Trump’s for nearly two decades, and was one of his most prominent early supporters after it became clear that Mr. Trump was the leading contender for the Republican nomination in 2016. But after Mr. Trump’s election, he was dismissed as head of his transition team, and he did not end up with a job in the administration. People close to him said the only one he had been interested in was attorney general.

Still, Mr. Christie has remained close to the president and was considered one of the top prospects for replacing Mr. Kelly, who is scheduled to leave the White House in a few weeks. Mr. Christie, according to several White House officials, had some of the key characteristics of what Mr. Trump wants in a chief of staff. Paramount among them is personal chemistry with the president, who has made clear that he wants someone he personally likes for his third chief of staff.

Mr. Trump’s first choice to replace Mr. Kelly, Nick Ayers, a Georgia political operative who is now Vice President Mike Pence’s chief of staff, turned down an offer from the president over the weekend and plans to leave the administration. Representative Mark Meadows of North Carolina, who said publicly that he was interested in the job, was ruled out by the White House on Tuesday.

For Mr. Christie, who has two children in high school, the job would have been a personal sacrifice.

Mr. Christie has a memoir coming out next year, which is expected to touch on his tenure as a federal prosecutor in New Jersey. During that time, he prosecuted Charles Kushner, the father of Jared Kushner, Mr. Trump’s son-in-law, a task that has since complicated his relationship with the Trump family.