The US secretary of state, John Kerry, has stayed on in Vienna for a second day of talks with his Iranian counterpart in an attempt to break an impasse in international negotiations on Iran's nuclear programme.

Kerry entered what the US described as a "serious and potentially lengthy conversation" with Mohammad Javad Zarif, the Iranian foreign minister, in a Vienna hotel late on Monday morning, continuing talks adjourned on Sunday night. The discussions come six days before a deadline for completing a comprehensive deal between Iran and six world powers on the future size and shape of an Iranian programme, and at a moment when the talks are deadlocked on a range of issues.

The most serious obstacle is a fundamental agreement on whether Iran should have an industrial-scale uranium enrichment capacity, enabling it to be self-sufficient in nuclear fuel for its nuclear power station at Bushehr, or to retain only a research-scale facility and rely on continued fuel imports from Russia, which built the Bushehr reactor.

Diplomats said Kerry and Zarif were looking for a compromise that could lead to an agreement by Sunday, or would justify agreeing an extension to the talks in anticipation of a deal.

"There are a lot of issues to discuss as part of these nuclear talks, especially because 20 July is coming up so soon," said a senior state department official. "One of the goals of Secretary Kerry coming to Vienna was to have in-depth discussions with Foreign Minister Zarif to gauge Iran's willingness to make the critical choices it needs to make. That's a pretty serious and potentially lengthy conversation. The secretary will take the time necessary to have that discussion, and that's why they'll be meeting again today, to see if progress can be made."

Zarif told reporters that the Sunday night talks had made "some important headway".

Kerry arrived in Vienna soon after Iran's supreme leader, Ali Khamenei, appeared to harden Iranian positions, publicly stating the country's demand for an enrichment capacity more than 10 times bigger than its current size.

Western diplomats have played down the idea of an extension to the talks, arguing it would be pointless without more signs of flexibility from Tehran.

A senior European diplomat said: "We don't believe an extension will do any good. It's now or never."

Kerry and Zarif are thought to be discussing the possibility for a phased agreement, in which Iranian enrichment capacity is restricted to a few thousand centrifuges over the next few years, with the prospect of later expansion based on demonstrable need and improved Iranian transparency about the country's nuclear activities.

The foreign ministers of the UK, France and Germany flew in to Vienna to take part in the talks on Sunday, but left the same evening, leaving Kerry and Zarif to discuss the core issues.