But his latest jab continued a complicated history with the magazine. In 2011, he said Time had “lost all credibility” for not naming him to its Top 100, and in 2015, he complained that he was passed over in favor of Chancellor Angela Merkel of Germany despite being “the big favorite.”

Yet he appeared to aspire to be on the magazine’s cover, with a fake 2009 cover story once hanging near the entrance of Mar-a-Lago, the Florida estate where he is spending his vacation, and in many of his other golf clubs, according to a Washington Post article in June. (A White House spokeswoman declined at the time to say whether Mr. Trump had known that the cover wasn’t real.)

At the time of Mr. Trump’s tweet on Friday, an online readers poll on whom the magazine should select showed Mr. Trump in a three-way tie for second and trailing Crown Prince Mohammed bin Salman of Saudi Arabia, who had 21 percent of the vote. The recipient of the title, who is ultimately decided by Time’s editors, will be announced on Dec. 6.

A spokeswoman for the magazine directed reporters back to Twitter. “The President is incorrect about how we choose Person of the Year,” a message from the magazine’s account said. “TIME does not comment on our choice until publication, which is December 6.”

In between his bookends of institutional critique, the president fit in a few moments of international productivity on Friday.