Zarif made the demand in a telephone conversation with his British counterpart Dominic Raab, the Ministry of Foreign Affairs said in an online report on Sunday.
The two diplomats discussed the latest developments related to the deal, officially known as the Joint Comprehensive Plan of Action (JCPOA) in the call, held at Raab's request, Xinhua news agency reported citing the report as saying.
Zarif reiterated Tehran's position that Iran will return to its JCPOA obligations after the "illegal and unjust" sanctions imposed by the US are "completely and definitively" lifted and verified by the Islamic Republic.
Raab said, according to the report, that his country will make efforts for the negotiations to bear fruits.
Parties to the Iran nuclear deal have agreed to convene in Austria on Tuesday to discuss ways to bring the deal back to life.
US State Department spokesman Ned Price confirmed that American representatives will attend the meeting, adding that the country remains "open to" direct talks with Iran.
But, Iran has ruled out any "direct or indirect" contacts with US representatives in Vienna.
The two countries are in a standoff over reviving the nuclear deal.
The Joe Biden administration has said that if Iran returns to full compliance with the JCPOA, Washington would do the same.
But Iran insists its full compliance will only take place once US sanctions are removed.
The deal was reached in 2015 between Iran and the US, the UK, Russia, France, China, Germany and the European Union.
Iran gradually stopped implementing parts of its JCPOA commitments from May 2019, one year after the administration of former US President Donald Trump unilaterally abandoned the agreement and re-imposed sanctions against Iran.