Mr. Trump’s aides have grown concerned that the president — who has said that “everyone thinks” he deserves a Nobel Peace Prize for his efforts — has signaled that he wants the summit meeting too much. The aides also worry that Mr. Kim, sensing the president’s eagerness, is prepared to offer assurances that will fade over time.

Moreover, Mr. Trump’s decision this month to withdraw from the 2015 Iran nuclear deal raises the stakes for the North Korea negotiation. If he emerges with anything less than what President Barack Obama got, which in Iran included the verified shipment of 97 percent of all nuclear material out of the country, it will be hard for Mr. Trump to convince anyone other than his base that the negotiation was a success.

The aides are also concerned about what kind of grasp Mr. Trump has on the details of the North Korea program, and what he must insist upon as the key components of denuclearization. Mr. Moon and his aides reported that Mr. Kim seemed highly conversant with all elements of the program when the two men met, and Secretary of State Mike Pompeo has made similar comments about Mr. Kim, based on his two meetings with him in Pyongyang, the North’s capital.