Both sides, seeking to end the standoff over Iran's nuclear programme after a decade of rising tensions, stressed however that the talks were detailed, serious and constructive.
Raising the pressure, US Senate Majority Leader Harry Reid said in Washington lawmakers would move to impose new sanctions on Iran in December if there is no deal.
The United States, China, Russia, Britain, France and Germany — the so-called P5+1 — want Iran to freeze parts of its nuclear programme for six months in return for relief from painful sanctions.
This hoped-for "first phase" deal would build trust and ease tensions while Iran and the six powers hammer out a final accord that ends once and for all fears that Tehran will get an atomic bomb.
Numerous attempts to resolve the standoff have failed over the last decade, but the election this year of Rouhani as Iranian president has raised hopes that this time a deal can be struck.
The proposed accord includes suspending uranium enrichment to 20 percent purity — close to weapons-grade — asl well as the removal of uranium stockpiles and tighter UN inspections.
For Israel, which refuses to rule out military action against Iran, the proposal does not go far enough.
Israeli Prime Minister Benjamin Netanyahu wants the complete and permanent dismantling of all of Iran's nuclear facilities, not just those enriching to 20 percent.
"Yesterday, Iran's supreme leader, (Ayatollah Ali) Khamenei, said 'death to America, death to Israel', he said that Jews are not human beings," Netanyahu said in Moscow.
"The Iranians deny our past and speak time afer time of their commitment to wipe the state of Israel off the map."
But Daryl Kimball from the Arms Control Association said a complete dismantling might have been possible in 2005 when Iran had fewer than 300 centrifuges at one site.
"But it is not realistic now that Iran has 19,000 installed and 10,000 operating centrifuges at two sites," he said.
'Points of difference'
Iran's lead negotiator said after a second meeting between Foreign Minister Mohammad Jarad Zarif and P5+1 counterpart Catherine Ashton in Geneva that a deal was not yet in view.
But Abbas Araqchi added that the talks had been "serious" and had focused on remaining "points of difference."
Ashton's spokesman said that the talks had been "useful", with "intensive work continuing" and negotiations to resume later.
Similar talks two weeks ago came close to succeeding, prompting US Secretary of State John Kerry and other foreign ministers to jet into Geneva ready to sign a deal.
Rumours were flying around Geneva on Thursday that Kerry and other top diplomats were gearing up to return to Switzerland but this has not been confirmed officially.
Last time they failed to agree after France insisted that a proposed deal did not go far enough in securing guarantees on Iran's uranium enrichment.
But French Foreign Minister Laurent Fabius said Thursday that the text being debated with the Iranians on Thursday was "supported by all six" world powers.
"This deal will only be possible if it has a firm base," Fabius told France 2 television.
Frozen oil billions
Western powers say that the relief from painful sanctions that Iran would get in a deal would be minor, and that the main oil and banking sanctions would stay during this period.
Estimates on how much this is worth have varied wildly between $6 and $50 billion but the US ambassador to the United Nations Samantha Power told CNN Thursday "the lower number is closer to what we're talking about."
The relief is likely to come from unblocking some of the $100 billion in Iranian reserves from oil sales, most of it frozen in bank accounts around the world.
If Rouhani, meanwhile, fails to secure quick and substantial relief from the sanctions, he risks losing the support of arch-conservatives and the supreme leader, experts say.
Another sticking point is Iran's insistence that its "right" to enrich uranium is recognized by the P5+1, even though this is not explicitly set out in the nuclear Non-Proliferation Treaty.