Such an accord is aimed at killing off once and for all worries that Iran might develop nuclear weapons under the guise of its civilian programme, and silence talk of war.
Iran denies seeking the bomb and wants the lifting of all UN and Western sanctions, which have caused it major economic problems.
The five permanent members of the UN Security Council plus Germany have been negotiating almost constantly for months, but the talks have come up against major problems -- as expected.
The European ministers left today however saying no breakthrough had been made, although Kerry remained for further discussions with his Iranian counterpart Mohammad Javad Zarif.
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Russia and China sent only lower-ranking officials, with Chinese Deputy Foreign Minister Li Baodong urging both sides "to show flexibility".
Kerry said on arrival that "very significant gaps" remained, while Iranian negotiator Abbas Araqchi said that on all the important issues, no narrowing of positions was evident.
Germany's Foreign Minister Frank-Walter Steinmeier, who like the others held one-on-one talks with Zarif, was the most downbeat, warning that "the ball is in Iran's court".
Britain's William Hague said that no "decisive breakthrough" was achieved and that there remained a "huge gap" on the key issue of uranium enrichment -- an activity that can produce fuel for the country's sole nuclear plant or, if further enriched, the matter for an atomic bomb.
The six powers want Iran to reduce dramatically the scope of its enrichment programme, while Tehran wants to expand it.
Israel, the Middle East's sole if undeclared nuclear weapons state and which together with Washington has refused to rule out military action, is opposed to any enrichment by Iran.
"It would be a disaster for the United States and for everyone else," he told Fox News, adding that "a bad deal is actually worse than no deal".