The buff star of action flicks including The Bourne Identity has been dating Barosso for 12 months since he met the aspiring interior designer at the Miami bar where she worked.
Since then, Barosso has travelled the world with the 34-year-old actor, staying with him in Europe during the filming of Ocean’s 12 and even travelling Down Under late last year for the premiere of The Bourne Supremecy.

A friend told American gossip mag Us Weekly the pair were engaged but “it’s all very hush, hush”.

Damon’s spokesperson has refused to confirm or deny the reports.

Damon has been linked to several Hollywood stars in the past including Winona Ryder and, most famously, Minnie Driver, who heard news of the couple’s break-up via the Oprah Winfrey show.

But, after the media circus surounding good pal Ben Affleck and Jennifer Lopez, Damon told the Sunday Telegraph Sunday Magazine last year that he was sworn off celebrity romances and now only dates “civilians”.