Are Ryan Gosling and Michelle Williams a Couple Off-Screen?

Their new movie Blue Valentine earned great reviews so far at the Cannes Film Festival – but costars Michelle Williams and Ryan Gosling have everyone on Croisette talking about something else: whether or not they're a real-life couple.

In the drama, Williams and Gosling, both 29, play a couple trying to prevent their marriage from falling apart. But the stars looked anything but troubled at Tuesday's photocall, where the actor squeezed Williams's hand affectionately.

Later, just before their big premiere, Williams clasped Gosling's hand on the red carpet, and the two remained hand-in-hand while walking into the screening. They were equally cozy at a rooftop afterparty at the Palais Stephanie hotel, where Gosling kept his arm firmly around Williams's shoulder while partygoers sipped cocktails around a pool with views of the lit-up yachts in the harbor.

So, are they an item?

"He is cozy with everyone!" Gosling's rep tells PEOPLE.

A source close to the film's producers believes they were dating. "You know actors who meet on movie sets," he said. "It can be the most romantic place in the world."

This isn't the first time that rumors about their relationship have surfaced. At the Sundance Film Festival in January, where Blue Valentine initially premiered, Gosling, who brought his mom as his date, and Williams showed obvious chemistry at a dinner for the cast.

They also danced with Faith Wladyka, who plays their daughter, and took turns tubing down the snow hill.