“You can’t solve problems if you’re not talking about them,” Mr. Huntsman said Thursday during a briefing for reporters about the meeting. Mr. Trump thinks a better relationship between Russia and the United States would be good for both, he said, “but the ball really is in Russia’s court, and the president will continue to hold Russia accountable for its malign activity.”

He said that Mr. Trump saw a one-on-one exchange as vital to starting a dialogue with Mr. Putin. But Mr. Huntsman played down the prospects for any breakthroughs, saying, “The fact that we’re having a summit at this level at this time in history is a deliverable in and of itself.”

Still, there are serious issues to be discussed. Mr. Trump is likely to raise the prospect of extending the New Start arms reduction treaty, Mr. Huntsman said, and to urge Russia to return to compliance with the Intermediate-Range Nuclear Forces Treaty, after Moscow tested and deployed a cruise missile prohibited under the pact.

Yet Mr. Trump’s advisers concede they cannot be certain of what he will choose to say or do.

Mr. Huntsman said the United States and Russia have yet to hold the kind of “direct conversations, across-the-table conversations — about things like election meddling and malign activity — that really do need to take place.” But he said it would be up to Mr. Trump to choose how he wished to broach those topics with Mr. Putin.

The Trump administration over all has been far tougher on Russia than the president himself has been willing to be, at least in public. Mr. Trump in recent weeks has spoken openly about readmitting Russia into the Group of 7, the club of industrialized democracies from which it was expelled in 2014 after Mr. Putin annexed Crimea.

In the past, Mr. Trump’s informal meetings with Mr. Putin appear to have played down points of friction, particularly over the topic of Russia’s election meddling, which Mr. Putin has vehemently denied. Mr. Trump has often contended that talk of Russian election meddling is an overblown accusation meant to undercut his legitimacy, even though such statements are contrary to the findings of the United States intelligence community, which concluded that it was a serious effort by Mr. Putin himself.

When the two met on the sidelines of the Group of 20 summit meeting about a year ago in Hamburg, Germany, American officials said Mr. Trump confronted Mr. Putin about the election meddling, but the Russian president denied it and both agreed it was time to move beyond the matter. Russian officials said Mr. Trump had “accepted” Mr. Putin’s denial and told him that certain people in the United States were “exaggerating” the issue.

Mr. Trump said that he had raised the subject with Mr. Putin again during a discussion on the sidelines of the Asia-Pacific Economic Cooperation gathering in November in Danang, Vietnam, and that he believed the Russian president’s denials were sincere. Mr. Trump said then that the continued focus on election meddling was “insulting” to Mr. Putin, and called the special counsel investigation about it a Democratic “hit job” that was hindering progress between the two countries on other issues.