On Tuesday, Marshall Billingslea, Mr. Trump’s lead negotiator, announced that the two leaders had an “agreement in principle, at the highest levels of our two governments, to extend the treaty.” Mr. Billingslea described an added “gentleman’s agreement” to cap each country’s stockpile of weapons not currently deployed on missiles, submarines or bombers. Details needed to be worked out, he cautioned, including the tricky work of verifying compliance.

It sounded like a promising solution, for a few hours.

Then the Russian deputy foreign minister, Sergei A. Ryabkov, shot back that this was a figment of someone’s election-season imagination. “Washington is describing what is desired, not what is real,” he said in a statement. For example, he said, Moscow would not freeze the number of tactical weapons it possesses.

With less than three weeks to Election Day, it seems no agreement is in the offing, and Trump administration officials are saying that, after the election, the price will go up. Former Vice President Joseph R. Biden Jr., the Democratic nominee who was involved in the negotiation of the original agreement in 2010, has indicated that, if elected, he will agree to a straightforward, immediate extension of the accord for five years, the maximum allowed under the current terms, and then work to expand its scope.