Two allies of Mr. Mattis sent emails to associates on Friday notifying them that the retired general had closed the door on a campaign. William Kristol, editor of the conservative Weekly Standard magazine, said Mr. Mattis had decided “after much consideration” not to proceed.

“The thoughtfulness and patriotism — and for that matter, the modesty — Jim showed as he reflected on this decision make me more convinced than ever that he would have made a truly admirable president, and also a good candidate,” Mr. Kristol wrote. “But it’s not to be. So we won’t have a President Mattis.”…

In private, Mr. Mattis, 65, was receptive to political overtures: During a visit to Washington last Friday, he met with a small group of strategists supportive of his entry into the race and discussed the election, according to people familiar with the meeting, who spoke on condition of anonymity about the private session.