“Is it possible in the end we won’t reach an agreement?” Mr. Kerry said in a news conference hours before the deadline on Monday night. “Absolutely.”

“These talks are not going to get easier just because we extend them,” Mr. Kerry acknowledged. “They are tough, they’ve been tough, and they are going to stay tough.”

Officials said little about the new approaches they were exploring with Iran, other than to indicate that “experts” — presumably at the Energy Department’s national laboratories — would be studying them to see if they, in combination with other steps, would result in at least a year’s warning if Iran raced for a weapon. That is the standard the United States has set.

The steps could involve a combination of Iranian commitments to ship some of its nuclear stockpile to Russia, efforts to disconnect some of the country’s centrifuges in ways that would take considerable time to reverse, and limits on output that could be verified by international inspectors. “It’s a lot of moving parts,” said one European diplomat involved in the discussions, “and the question is what it adds up to.”

Mr. Zarif said late Monday that the new ideas did not represent a proposal by either side but rather, “We have a compilation of ideas that each side has given the other.” He said he would not violate the secrecy of the negotiating room and, like Mr. Kerry, argued that leaks emerging from the discussions — and there have been many in the talks, involving six different nations — make it far harder to reach a deal.

Mr. Kerry went out of his way to compliment the lead Iranian negotiator, Mr. Zarif, who American officials have described as a creative diplomat who is forced to navigate Iran’s treacherous politics and is uncertain how far the country’s supreme leader will let him go. He and Mr. Rouhani came to power promising an end to sanctions that have reduced Iran’s oil revenue by roughly 60 percent, crashed its currency and made overseas financial transactions almost impossible.