Adding diplomatic weight to the negotiations, Russian Foreign Minister Sergei Lavrov had arrived in Geneva, where his US, British, French and German counterparts rushed on Friday hoping to seal a breakthrough.
The hoped-for agreement -- seen as a first step ahead of further talks on a final deal -- could see Tehran freeze its nuclear efforts for as long as six months in exchange for some relief from the sanctions that have battered its economy.
But hopes were nonetheless high that a breakthrough first-stage agreement was in the works in the decade-old standoff.
"These negotiations have made very good progress and continue to make progress," British Foreign Secretary William Hague told journalists, adding though that it was "too early" to be sure a deal could be reached on Saturday.
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"There is now a real concentration on these negotiations so we have to do everything we can to seize the moment and seize the opportunity to reach a deal that has eluded the world," he said.
"There is an initial draft that we do not accept... I have no certainty that we can finish up" at this stage, Fabius told France Inter radio before talks started on Saturday.
In a series of meetings early today, Kerry and the European ministers met with EU diplomatic chief Catherine Ashton, who has represented the six world powers at the talks.
Fabius then entered separate talks with Ashton and Iranian Foreign Minister Mohammad Javad Zarif.
If some sort of agreement is reached, it would be a breakthrough after a decade of negotiations between Iran and the P5+1 group comprising the United States, Britain, France, Germany, Russia and China.
Reports say the proposed deal could see Tehran stop enriching uranium to 20 percent, which is just a few technical steps from weapons grade, reduce existing stockpiles and agree not to activate its plutonium reactor at Arak.