By Parisa Hafezi and John Irish

DUBAI/CALGARY, Canada (Reuters) -U.S. special envoy Steve Witkoff and Iranian Foreign Minister Abbas Araqchi have spoken by phone several times since Israel began its strikes on Iran last week, in a bid to find a diplomatic end to the crisis, three diplomats told Reuters.

According to the diplomats, who asked not to be identified due to the sensitivity of the matter, Araqchi said Tehran would not return to negotiations unless Israel stopped the attacks, which began on June 13.

They said the talks included a brief discussion of a U.S. proposal given to Iran at the end of May that aims to create a regional consortium that would enrich uranium outside of Iran, an offer Tehran has so far rejected.

U.S. and Iranians officials did not immediately respond to a Reuters request for comment on the matter.

This week's phone discussions were the most substantive direct talks since the two began negotiations in April. On those occasions, in Oman and Italy, the two men exchanged brief words when they encountered each other after indirect talks were held.

A regional diplomat close to Tehran said Araqchi had told Witkoff that Tehran "could show flexibility in the nuclear issue" if Washington pressured Israel to end the war.

A European diplomat said: "Araqchi told Witkoff Iran was ready to come back to nuclear talks, but it could not if Israel continued its bombing."

Other than brief encounters after five rounds of indirect talks since April to discuss Iran's decades-old nuclear dispute, Araqchi and Witkoff had not previously held direct contacts.

A second regional diplomat who spoke to Reuters said "the (first) call was initiated by Washington, which also proposed a new offer" to overcome the deadlock over clashing red lines.

URANIUM ENRICHMENT

U.S. President Donald Trump wants Tehran to end uranium enrichment on its soil, while Iran's Supreme Leader Ayatollah Ali Khamenei has said Tehran's right to enrichment is non-negotiable.

Trump has been keeping his cards close to his chest over whether he will order U.S. forces to join Israel's bombing campaign that it says aims to destroy Iran's nuclear programme and ballistic capabilities. But Trump offered a glimmer of hope that diplomacy could resume, saying Iranian officials wanted to come to Washington for a meeting.

He rebuffed President Emmanuel Macron earlier this week when the French leader said Trump had told G7 leaders at a summit in Canada that the United States had made an offer to get a ceasefire and then kickstart broader discussions.

European officials have been coordinating with U.S. Secretary of State Marco Rubio, who was also at the G7 summit.

Britain, France and Germany, known as the E3 and party to a 2015 nuclear deal between world powers and Iran, held a ministerial call with Araqchi on Sunday. The three countries and the European Union are set to meet him in Geneva on Friday, Iranian Foreign Ministry spokesman Esmaeil Baghaei and an EU official said.

Earlier in the week, both Rubio and Araqchi told the Europeans in separate talks about a possible diplomatic initiative, three diplomats said.

A senior European diplomat said what emerged at the G7 was that Trump wanted the operations to end very quickly and that he wanted the Iranians to talk to him, while making clear that they had to accept his demands if they wanted the war to end.

Given the Israeli strikes and Trump's rhetoric, diplomats said Iran was in no position to hold public talks with the U.S., but that a meeting with the Europeans as a link to try and advance diplomacy was deemed more realistic for Tehran.

(Writing by John Irish and Parisa HafeziEditing by Gareth Jones)