Other Russian officials have for months spoken of plots in America against Mr. Trump, echoing the American leader’s own criticisms of the special counsel investigation into Russian election interference and possible collusion with members of Mr. Trump’s campaign team as the work of the “deep state.”

“We see that there are forces in the United States that put their own group and narrow partisan interests above the national ones,” Mr. Putin said. “Our renowned satirists once wrote very well about such people: ‘Pathetic, paltry people.’ But this is not so in this particular case: These people are not pathetic and not paltry. On the contrary, they are quite powerful and strong if they can, excuse my crudeness, force-feed millions of their people various stories that are hard to digest in normal logic.”

In speaking of “pathetic, paltry people” in reference to opponents of Mr. Trump’s Russia policies, Mr. Putin appeared to be quoting from the book “The Twelve Chairs” by the Russian satirists Ilya Ilf and Yevgeny Petrov, using a derisive phrase that is familiar to readers of Russian literature.

Mr. Putin has also responded to criticism stemming from the news conference after the summit meeting, issuing a clarification of his own comments.

He suggested on Monday that Moscow might allow American law enforcement agents to question 12 Russian men identified by the special prosecutor, Robert S. Mueller III, as military intelligence officers who hacked the Democratic National Committee and other targets during the 2016 election, if the United States allowed Russia to question American intelligence agents and others suspected of illegal activity in Russia.

Mr. Trump called it “an incredible offer.”

Back in Washington, the White House press secretary, Sarah Huckabee Sanders, said on Wednesday that the idea was under consideration, though a State Department spokeswoman, Heather Nauert, called the idea “absolutely absurd.” On Thursday, Ms. Sanders said Mr. Trump disagreed with the proposal.

As an example of how such an arrangement could be helpful, Mr. Putin had said Russian law enforcement agents wanted to question associates of a onetime investor in Russia, William F. Browder, who he said had violated Russian tax laws and donated $400 million of the gains to the Democratic Party. Mr. Browder denied violating Russian tax laws.