Mr. Pompeo’s warning is the first known rebuke from a senior American official to Russia over the bounties program, and it runs counter to President Trump’s insistence that the intelligence from U.S. government agencies over the matter is a “hoax.” The action indicates that Mr. Pompeo, who previously served as Mr. Trump’s C.I.A. director, believes the intelligence warranted a stern message.

Mr. Pompeo delivered the warning in a call on July 13 with the minister, Sergey V. Lavrov, choosing to do so during a conversation that, officially, was about an unrelated topic — the possibility of a meeting of the five permanent members of the United Nations Security Council, the U.S. officials said in the past week.

The secretary of state did not explicitly point to the covert bounties scheme organized by a Russian military intelligence unit that was first reported in late June by The New York Times, most likely because the details of what American intelligence has learned and how it gathered the information remain classified, one of the officials said. In public, Mr. Pompeo has carefully avoided answering direct questions about American intelligence on the Russian bounties. But late last month in congressional testimony, he said broadly that he had raised with Mr. Lavrov “all of the issues” that put American interests at risk.