President Obama has said the Iranian nuclear crisis could be resolved by bilateral negotiations between Washington and Tehran.

In the course of the last of three presidential debates – which chiefly served to underline the striking similarities in the foreign policies of the president and his challenger, Mitt Romney – Obama appeared to open a new route towards a negotiated settlement to the worsening impasse over Iran's nuclear programme.

For the past few years, talks with Iran have been handled by a group of six major powers: the five permanent members of the security council together with Germany. The US has had occasional meetings with Iranian officials alongside multilateral talks but the Iranians, hamstrung by deep internal divisions in Tehran, have shied away from such public encounters over the past three years.

During Monday night's debate in Florida, two weeks before what is expected to be a close election, Obama dismissed a New York Times report over the weekend that the US and Iran were exploring the possibility of holding direct bilateral nuclear negotiations after the election. But a few minutes later, he appeared to contradict himself, in what was possibly an unguarded remark made out of irritation that Romney had taken to echoing many of his administration's policies and presenting them as his own.

"I'm pleased that you now are endorsing our policy of applying diplomatic pressure and potentially having bilateral discussions with the Iranians to end their nuclear programme," he said, although Romney had made no mention of such discussions.

While the White House has rejected the New York Times report, it has not specifically denied that American and Iranian officials have been holding secret meetings in parallel to the public multilateral negotiations, since soon after Obama came to office in 2009.

Mark Fitzpatrick, a former state department expert on proliferation at the International Institute for Strategic Studies, said: "I have been hearing for some time that they had been having private discussions, and now it is starting to become public."

Western officials say that within a few weeks of the US election, a new round of talks are expected between the six powers and Iran at which the offer put on the table in return for curbs on Iran's uranium enrichment would be "reformulated".

The six powers (US, UK, France, Russia, China and Germany) will make it clearer to Iran that relief from the current severe sanctions regime will be available if Iran stops producing 20%-enriched uranium – a particular proliferation concern as it could easily turned into weapons-grade uranium should Iran take the decision to make a bomb.

US officials have said they expect that meeting to go ahead, but President Obama signalled on Monday night it could be a springboard to a new bilateral negotiating track to run separately or in tandem with the broader talks.

"There will be a new round of talks before the end of November, and I think what the Iranians have agreed to is to meet bilaterally with US officials at the margins of those talks," Fitzpatrick said.

"Ultimately, this is something that has to be resolved between the US and Iran. They are the two main protagonists. Whatever they agree to, the others will go along with."

The prospect of a new phase in the negotiations possibly opens up a significant difference in outcomes in foreign policy as a result of the 6 November elections, where otherwise the differences are relatively minor.

It is not at all clear whether Romney would enter such talks without a full policy review, and his close relationship with the Israeli prime minister, Binyamin Netanyahu, suggests he would pay more heed to Israeli reservations about such negotiations, which are likely, if successful, to leave Iran with a limited right to uranium enrichment. It is also far from assured that, if presented with such an opportunity, Iran's elderly and infirm supreme leader, Ali Khamenei, would have the confidence to embrace it.