Negotiators from Iran and six world powers edged towards a nuclear deal on Sunday as both sides mounted a final diplomatic push to seal what could be a transformative agreement.

Though the official deadline for negotiations — the end of June — has passed, diplomats are closer than ever to agreeing the terms of a deal that would place long-term curbs on Iran’s atomic programme in return for an end to crippling international sanctions against the Islamic Republic.

A deal has the potential not only to transform Iran’s economy, but to reshape the geopolitics of the Middle East.

In the US, the process has become the most important foreign policy priority for Barack Obama’s administration, even though many of Washington’s staunchest regional allies — notably Israel and Saudi Arabia — are implacably opposed to the negotiations.

US diplomats are seeking a deal by Tuesday at the latest, while a flurry of activity over the weekend raised the possibility of an agreement sooner.

The White House needs to present the terms of the accord to Congress before Thursday, when legislation kicks in giving US politicians two months to scrutinise any deal — offering hawks greater opportunity to unpick it.

Speaking on Sunday outside the Coburg Palace in Vienna, where the talks are taking place, John Kerry, US secretary of state, said: “We are not where we need to be. But we are closer than we have ever been.”

The so-called P5+1 — the five permanent members of the UN Security Council plus Germany — agreed the blueprint for an accord in Lausanne, Switzerland, in April. Three months of technical discussions since then have fallen short of fully fleshing out the agreement and solving the outstanding issues.

Mr Kerry has spent six consecutive days in head-to-head negotiations with Mohammad Javad Zarif, Iran’s foreign minister.

The final bout of negotiations between Mr Kerry and Mr Zarif began last week, with senior western diplomats initially saying Iran was attempting to row back on some of the issues resolved in the Lausanne blueprint.

The deadlock appeared to break after Mr Zarif flew to Tehran and returned with Hossein Fereydoun, brother and close aid to President Hassan Rouhani, and Ali Akbar Salehi, Iran’s nuclear chief. Their presence was seen by many as a sign that progress on outstanding technical issue was being made.

Of the three sticking points, two appear close to resolution, according to diplomats.

First, the issue of sanctions relief, and the phasing of it, has been largely agreed upon, according to two P5+1 negotiators. The deal will involve an immediate easing of financial sanctions and an end to international sanctions imposed by the UN Security Council. A compromise has also been found over the issue of an automatic “snapback” mechanism, under which sanctions would be reimposed if Iran reneged on the deal. A committee of specialists will oversee compliance and report to the security council.

Second, issues surrounding “possible military dimensions” to Tehran’s nuclear programme have also come close to resolution. Yukiya Amano, head of the International Atomic Energy Agency, said he expected outstanding questions in this area — a requirement of the P5+1 — to be resolved by the end of the year.

On the third unresolved issue, however, Iran has resisted proposals that would give international inspectors access to sensitive military sites to check compliance after the deal comes into force.

Ayatollah Ali Khamenei, Iran’s supreme leader, has repeatedly insisted he would not give ground on this issue, although the P5+1 had hoped this was grandstanding to extract a better deal.

Iranians, meanwhile, are on tenterhooks awaiting the outcome of the Vienna talks. Many see the deal as having the potential to change their lives by rejuvenating their economy and opening the country up to the world after decades of mistrust.

“My optimism is still more than my pessimism, but officials are vague and contradict each other, which is worrying,” said Zahra, a housewife.

Shiva, a secretary, added: “I hope prices of basic commodities drop after a deal because high inflation is ruining our lives. If there’s no deal, we should probably go and die.”

Abbas Araghchi, a member of Iran’s negotiating team, told state television over the weekend that a deal was close, although differences remained over the development of advanced centrifuges, the timetable for lifting sanctions and restrictions on Iran’s uranium enrichment activities.

He added that Iranian negotiators would rather return to Tehran “empty-handed” than strike a bad deal — in comments that could be to allay concerns of Iran’s hardliners, who oppose a deal, which they say is already stained with too many concessions.