Sarah F. Cliffe, a former United Nations assistant secretary general who is now director of the Center on International Cooperation, a research organization at New York University, said she expected a reprise of the tensions that erupted between the United States and the United Nations during the administration of President George W. Bush. John R. Bolton, who was ambassador to the United Nations under Mr. Bush, once said the United Nations would be more effective without its top 10 floors, where its senior leaders have their offices. Mr. Bolton is one of Mr. Trump’s many would-be candidates for secretary of state.

But Ms. Cliffe said Mr. Trump may also find the United Nations useful.

“He prides himself on making deals,” she said. “The U.N. is the forum where countries make deals in their own national interests but that also does some collective good.”

His own comments about the United Nations are difficult to parse. At a 2005 Senate hearing about planned renovations of the United Nations headquarters, he described himself as “a big fan” and said that “the concept of the United Nations and the fact that the United Nations is in New York is very important to me and very important to the world as far as I am concerned.”

As one of Manhattan’s pre-eminent builders, he also offered to handle the renovation at half the price. In 2012, he complained on Twitter about the marble behind the speaker’s lectern at the General Assembly hall, claiming he could build it better.