President Trump plans to invite President Vladimir V. Putin of Russia to Washington for another meeting in the fall, officials said Thursday, even as Mr. Trump’s top advisers groped for details of what the two leaders discussed in their last meeting in Helsinki, Finland.

Mr. Trump’s director of national intelligence, Dan Coats, acknowledged frustration at being kept in the dark about the meeting, which included only the leaders and their respective interpreters. “If he had asked me how that ought to be conducted,” Mr. Coats said at a security conference in Aspen, Colo., “I would have suggested a different way. It is what it is.”

As the questions mounted, the White House rejected a proposal by Mr. Putin to question American citizens, including a former ambassador to Moscow, Michael A. McFaul, in return for giving the United States access to 12 Russian military intelligence officers indicted for their role in trying to sabotage the 2016 election.