But given the failure to close the wide gap that remains, neither leader seems inclined to end the negotiations — or to declare failure. The Iranians have reportedly declined to agree to dismantle a significant number of their centrifuges — the machines that enrich uranium at supersonic speed — just as Mr. Obama has refused to end the sanctions by a specific date, until he can measure Iran’s compliance.

And other issues are in dispute. One is how many years an agreement would last before Iran would be free, like any other signer of the Nuclear Nonproliferation Treaty, to develop as large a civilian nuclear infrastructure as it would like. Another is what kind of freedom inspectors would have to visit any location where they suspect nuclear-related work might be underway.

Image Iranian negotiators have expressed concern about President Obama’s ability to deliver an agreement that will survive contact with a skeptical Congress. Credit... Stephen Crowley/The New York Times

For weeks, the American team has sought to keep the pressure on the Iranians to make hard decisions in the talks by insisting that an extension was not on the table. “We are not talking about an extension,” Mr. Kerry insisted as recently as Thursday. “We are driving towards what we believe is the outline of an agreement that we think we can have.”

Republican and even some Democratic lawmakers have warned that they would press for additional sanctions if the American negotiators did not emerge from the latest round of talks with provisions that toughened the temporary agreement. That agreement freezes much of Iran’s nuclear program and is set to expire Monday evening.

The Obama administration already agreed to one extension in July, which it justified on the grounds that sufficient progress had been made to warrant continuing the talks until Nov. 24, the anniversary of an agreement for a temporary accord that froze some of Iran’s advances and required Iran to dilute a stockpile of fuel that the West feared could quickly be converted to weapons use.

A breakdown in talks, American and Iranian officials seem to agree, is in neither side’s interest.

Any extension that might be agreed to would presumably keep in place the freeze on much of Iran’s nuclear program and could be cited by the White House to make the case to Congress against additional sanctions. It would also enable the United States to argue to Prime Minister Benjamin Netanyahu of Israel — whom Mr. Kerry called on Saturday — that there is no need for military action because Iran’s nuclear threat is less than it was a year ago.