ROME (Reuters) - The nuclear deal reached between world powers and Iran could lead to better relations between Tehran and Washington if the United States apologised for past behaviour, Iranian President Hassan Rouhani was quoted as saying on Thursday.

The pragmatist president, who championed the July 14 deal, has pushed for closer engagement with the West since his 2013 landslide election win.

But Iran's top authority, Supreme Leader Ayatollah Ali Khamenei, has continued to rule out normalising ties with the "Great Satan", as he routinely calls the United States.

In an interview with Italy's Corriere della Sera newspaper, Rouhani suggested that the United States and Iran could open embassies in each other's capitals after decades of mutual hostility, but said Washington should apologise, without going into further detail.

"One day these embassies will re-open but what counts is behaviour and the Americans hold the key to this," Rouhani told the newspaper ahead of a trip to Italy this weekend, his first to a European capital.

"If they modify their policies, correct errors committed in these 37 years and apologise to the Iranian people, the situation will change and good things can happen."

Iran and Washington severed ties shortly after the 1979 Islamic revolution when radical students seized the U.S. embassy in Tehran and held 52 Americans hostage for over a year.

Relations came under further pressure in the last decade over Iran's nuclear ambitions.

Under the nuclear deal reached in July, Iran will curb its nuclear programme in exchange for an easing of sanctions on its economy. Tehran denied Western suspicions it wanted to develop an atomic bomb.

Khamenei, who has the final say on all state matters, gave his conditional approval to the deal with six world powers including the United States, but has warned against allowing any U.S. political or economic influence on Iran.

Rouhani said Washington would have to fulfil its part in the nuclear accord for relations to improve. The United States approved conditional sanctions waivers for Iran, though these will not take effect until Tehran has complied with the nuclear accord.

"The way this agreement is applied can have an impact on the future," Rouhani said in the interview.

"If it is well applied it can lay the foundation for fewer tensions with the United States, creating the conditions to open a new era. But if the Americans don't respect their part of the nuclear accord, then surely our relationship will remain as it has been in the past," he said.

Rouhani is due to see the Italian prime minister and business leaders during his Nov. 14-15 visit to Rome and will also hold talks with Pope Francis.

He will then fly to Paris for talks on Nov. 16-17.

(Writing by Philip Pullella; Editing by Sam Wilkin and Raissa Kasolowsky)