The diplomats said at least two other issues still needed final agreement: Iranian demands that a U.N. arms embargo be lifted and that any U.N. Security Council resolution approving the nuclear deal no longer describe Iran's nuclear activities as illegal.
They demanded anonymity because they are not authorized to discuss the negotiations publicly.
With few signs that Iranian or U.S. negotiators were prepared to give ground, the high-stakes game of brinksmanship looked set to force a fourth extension of talks since the current round began 17 days ago.
A temporary nuclear deal between world powers and Iran is set to expire at midnight Monday Vienna time (6 p.m. EDT), and diplomats had hoped to complete and announce a final agreement before day's end. But they warned there was no guarantee, and some said the talks could stretch into Tuesday despite little appetite from anyone for another extension.
"There should not be an extension in the talks," Iranian Foreign Minister Mohammad Javad Zarif said in remarks carried by Iran's official IRNA news agency. He called for diplomats to continue negotiating as long as necessary.
Grim-faced foreign ministers from the six countries negotiating with Iran declined to answer questions about another possible prolongation of discussions as they gathered for a group meeting at the 19th-century Viennese palace hosting the talks.
One diplomat described the latest delay as unexpected, saying negotiators expected to resolve the remaining disputes by late Sunday.
A deal would place long-term limits on Iran's nuclear program. The United States also wants to ensure the U.N.'s International Atomic Energy Agency has wide-ranging authority to investigate nuclear arms allegations after being stalemated for a decade.
IAEA chief Yukiya Amano returned from Tehran and talks with Iranian leaders early this month saying a new plan had been drawn up to aid his probe. And diplomats had cited progress within the Vienna negotiations on the issue. But the debate about how much access U.N. experts should be given has publicly continued, with top Iranian officials saying military sites and Iranian nuclear scientists would be off limits to IAEA monitors.
Iran insists it never worked on such weapons. It may be opposing any mention of concessions on the matter in the final, public documents describing the overall nuclear deal.
Iran's deputy foreign minister, Abbas Aragchi, told reporters in Vienna that the talks are at their "final breathtaking moments (but) certain issues still remain." He said he could not guarantee an agreement would be reached either Monday or Tuesday.
In Brussels, French President Francois Hollande said the sides are near agreement but "a gap" remains.
The foreign ministers of Russia and China, who had left the talks last week, both returned to the Austrian capital late Sunday. Most other foreign ministers of the seven nations at the table also were in Vienna by Monday, in place for any announcement.
"The foreign ministers are gathered to bring negotiations to a conclusion," said Wang Yi, China's top diplomat. "We believe there could not be further delay."
After more than two weeks of see-saw developments, including threats from both the United States and Iran to walk away, senior officials at the talks began to express optimism Sunday that a deal was within reach.
U.S. Secretary of State John Kerry said twice he was "hopeful" and met again with Zarif on Sunday evening. Afterward, foreign ministers and senior officials from the five permanent members of the U.N. Security Council and Germany held a group dinner.
A deal would culminate nearly a decade of wearying negotiations. The pact would impose verifiable limits on nuclear programs so that Tehran cannot modify them to produce weapons. Iran, in return, would get tens of billions of dollars in sanctions relief.
In the U.S., the Obama administration is under pressure from skeptical members of Congress and close American allies who feel the accord isn't stringent enough.
On Monday, Israeli Defense Minister Moshe Yaalon said a nuclear deal would force the Jewish state to "defend itself, by itself."