* Nuclear talks to resume June 18 in Moscow
* Iran says recognise its "nuclear rights" and deal possible
* US, EU demand access to suspect military site
By Fredrik Dahl and Marcus George
VIENNA/DUBAI, June 5 (Reuters) - If the world recognises Iran's "nuclear rights", negotiations aimed at easing a standoff with the West later this month could have a positive outcome, an adviser to Iran's supreme leader was quoted as saying on Tuesday.
But Washington said Iran had to move first to make its nuclear work compatible with international law and demanded it let U.N. inspectors into a military site that the West believes has been used for weapons-related nuclear research.
Iran has been under U.N. sanctions for years due to questions over its uranium enrichment - a process that yields fuel for power stations, Iran's stated goal, but also for bombs, if done to a much higher level.
Tehran says that as a member of the nuclear Non-Proliferation Treaty (NPT) it can develop a full nuclear fuel cycle, and, if this is recognised, talks with the permanent members of the U.N. Security Council - Britain, France, Russia, China, the United States - and Germany (the P5+1) can succeed.
"I hope the P5+1 group recognises Iran's inalienable nuclear right within the framework of the NPT and refrains from sitting on the sidelines," the official IRNA news agency quoted Ali Akbar Velayati, an aide to Ayatollah Ali Khamenei, as saying
Under the persistent threat of military strikes by Israel and ever tighter economic sanctions from the West, Iran returned to nuclear talks that had stalled in early 2011.
Diplomats say Iranian negotiators were more forthcoming at talks in Baghdad last month than in previous negotiations, and believe the supreme leader has given them a freer hand to explore a deal.
Talks are due to resume on June 18-19 in Moscow.
"By accepting Iran's right to use peaceful nuclear energy, the forthcoming talks in Moscow should reach a favourable result," Velayati said.
Washington said Iran must make the first move.
"We have long said we recognise Iran's right as a signatory to the NPT to the peaceful use of nuclear energy, but only after it comes into compliance with its international nuclear obligations," State Department spokesman Mark Toner said.
"What's at issue here is the fact they have not come into compliance," Toner told reporters. "If and when they come into compliance, at that point we'll address the possible civil use of nuclear energy."
A senior U.S. official said it was urgent that U.N. inspectors be allowed into the Parchin military site, near Tehran, where the West believes Iran conducted weapons-related research and is now covering up the evidence.
"It was clear from some of the images that were presented to us that further sanitisation efforts are ongoing at the site," said Robert Wood, acting head of the U.S. mission to the International Atomic Energy Agency, referring to satellite pictures the IAEA has shown to diplomats.
"If Iran has nothing to hide, why deny the agency access and carry out these apparent cleanup efforts?" Wood asked reporters at IAEA headquarters in Vienna.
He also called the continued expansion of Iran's underground Fordow nuclear facility a "serious provocation" and called for its immediate shutdown.
Iran has moved its most sensitive uranium enrichment activities to Fordow, located beneath a mountain, which would make it harder to destroy from the air.
Tehran has dismissed allegations about Parchin as "childish" and "ridiculous" and said it would consider allowing the IAEA to visit once a broader framework deal with the agency has been agreed on how to address inspectors' questions.
Iran and the IAEA will hold a new round of talks on Friday in Vienna in an attempt to reach an agreement on a so-called "structured approach" document which the U.N. agency hopes will enable it to resume its investigation in Iran.
Wood called on Tehran to sign the agreement at Friday's meeting but "history does not make me optimistic ... we have all seen this movie many times before with Iran."
The IAEA says the Islamic Republic has stonewalled its probe into the country's atomic activities for almost four years, and Western diplomats have voiced doubt that Iran will implement any accord that is reached.
Far from recognising Iran's "right" to enrich uranium, the European Union issued a statement at the IAEA saying Iran must suspend its enrichment activities and provide "early access" to Parchin.
Iran's ISNA news agency said Tehran had written twice to the P5+1 seeking preparatory meetings before talks in Moscow but had yet to hear back. (Additional reporting by Andrew Quinn in Washington and Zahra Hosseinian; Writing by Robin Pomeroy; Editing by Michael Roddy)