The two have struggled to make their long-distance relationship work and have called off their engagement, the Daily Mail reported.

“Uma and Arki were due to attend a gala together on April 8, but Uma showed up without her engagement ring or her fiance,” a source told the Daily Mail.

“Arki was scheduled to attend and it seemed clear there was only one reason why he didn't show – because he and Uma are no longer an item.”

The couple lead mostly separate lives – Thurman is based in New York, while Busson is based in London – and another source said this was the cause of their separation.

It's the second time the couple has called it quits.

The 43-year-old actress first began dating the French-born financier in 2007, and the couple became engaged a year later.

In 2009 Thurman told W that she had no plans to rush the wedding. “I'm happily engaged, but we haven't set a wedding up,” she told the magazine.

“I'm not in a rush to do anything. One day we'll do it, when we're ready.”

The actress foreshadowed the problems distance might cause, telling the magazine she could never leave New York.

“We can't live anywhere else because my children's father lives here,” she said.

That year Thurman returned her eight-carat diamond engagement ring to Busson, but the pair reconciled and began seeing each other again in 2011.

In 2012 they got engaged again and also had a daughter, who they named Rosalind Arusha Arkadina Altalune Florence (Luna) Thurman-Busson.

They both have children from different relationships, which is a reason neither Thurman nor Busson could move to another country full-time.

The Kill Bill star has two children from her marriage to actor Ethan Hawke: 15-year-old daughter Maya Ray, and 12-year-old son Levon.

The couple met on the set of sci-fi hit Gattaca and married in 1998. They divorced in 2005, and Thurman told Oprah Winfrey that the split from Hawke was “excruciating”.

“There's a lot of unfinished damage and baggage and stuff that's still in the way,” the actress told Winfrey about communicating with her ex.

“You bear two children with somebody – that's not a small thing – and then you can hardly talk to them.”

Thurman was also married to actor Gary Oldman from 1990 to 1992.

Busson, 51, has two children with his former partner Elle Macpherson, 16-year-old Arpad Flynn Alexander and 11-year-old Aurelius Cy Andrea.

Multimillionaire Busson and supermodel Macpherson started dating in 1996 and lived in London before separating in 2005.

At the time, the couple insisted the split was amicable, saying they remain “the greatest of friends”.

“We have had, and in many ways continue to have, a wonderful relationship, which has produced two beautiful children,” the couple said in a joint statement.