As it became clear in recent weeks that prison was unavoidable, Mr. Cohen shifted gears, moving to postpone his start date as long as possible. Mr. Cohen, who was sentenced in December, received a two-month delay to testify before Congress and recover from shoulder surgery. Last month, his lawyers asked congressional Democrats to seek another delay so he could help them sift through millions of his documents. Members of Congress declined the request.

Still, Mr. Cohen may be able to shave a few months off his sentence if prison officials determine that he has displayed good behavior while locked up. Mr. Cohen was also assigned to the prison he requested: the Federal Correctional Institution in Otisville, about 75 miles northwest of New York City.

The minimum-security camp there is appealing to Mr. Cohen for several reasons, including its proximity to his family in New York. It also offers a personal comfort for Mr. Cohen, the son of a Holocaust survivor, that is rare in the federal prison system: dozens of other Jewish inmates, as well as religious classes and weekly Shabbat services.

Mr. Cohen is not the only boldface name at the prison, which appeals to white-collar defendants of all religions who live in the New York area. Mike Sorrentino, who is known as The Situation and appeared in the MTV reality series “Jersey Shore,” is serving time there for tax evasion.

Mr. Cohen, who started working for the Trump Organization in 2007, first caught the real estate developer’s eye when he helped tamp down an uprising within the co-op board at a Trump property in Manhattan. He spent the next decade serving as his boss’s pit bull and mop-up man, handling personal issues for Mr. Trump and attacking reporters at his behest.

As congressional investigators and Mr. Mueller began encircling the president and those in his orbit, Mr. Cohen said publicly that he would “take a bullet” for Mr. Trump. But an F.B.I. raid on Mr. Cohen’s office, home and hotel room in April 2018 placed new strain on their relationship.

At that point, Mr. Cohen’s lawyer says, Mr. Trump’s lawyers “dangled” the possibility of a pardon; the president’s lawyers insist that they did not.