(Reuters) - U.S. Under Secretary Wendy Sherman and Tehran's Deputy Foreign Minister Abbas Araqchi met in Vienna on Thursday in the latest push to secure a final nuclear deal by a June 30 deadline, Iranian media said.

The latest round of nuclear diplomacy, being held in a luxury hotel in Austria's capital, kicked off with a bilateral meeting between Iran and the European Union on Wednesday.

Catherine Ray, spokeswoman for European Union foreign policy chief Federica Mogherini, said on Twitter that senior foreign policy officials from Iran, six world powers including Britain, China, France, Germany, Russia, and the United States and the EU will meet on Friday.

Western diplomats in New York said U.S. Secretary of State John Kerry and Iranian Foreign Minister Mohammad Javad Zarif were expected to meet next week on the sidelines of the five-year review conference on the nuclear Non-Proliferation Treaty.

One diplomat said the bilateral meeting at U.N. headquarters would focus on making progress toward a final nuclear agreement between Tehran and six world powers, though other issues could arise.

U.N. officials said Kerry and Zarif are expected to address the conference on Monday, the opening day of the month-long session. So far, the diplomats said, there are no plans for a full meeting of Iran and the six powers in New York during the April 27 to May 22 conference, though that could change.

Iran's nuclear negotiator Hamid Baidinejad told Iranian state television that "drafting the final deal has started," declining to give further details. Months ago both sides announced that a final draft had been started.

Iran and the other countries in the talks reached a tentative deal on April 2 and now aim to finalize the details by a self-imposed end of June deadline.

Under the framework agreement, Iran agreed to slash the number of uranium enrichment centrifuges it operates and to allow more intrusive inspections by the International Atomic Energy Agency (IAEA) in exchange for sanctions relief.

Diplomats need to iron out details about the timing of sanctions relief, the future of Iran's atomic research and development program, the exact nature of the IAEA's monitoring regime and what kind of uranium stockpile Tehran will be allowed to keep under any final deal.

The timing of sanctions relief was top of the agenda at Wednesday's talks.

Iran has said economic sanctions must be lifted as soon as any final deal is signed, while the United States wants a gradual lifting of restrictions.