The Drive star was photographed at Disneyland this weekend with his Place Beyond the Pines co-star Mendes, Us Weekly reports.
The two were seen holding hands and sharing a number of kisses while at the California amusement park.
"They were very playful," a witness told the website. "Eva was skipping around like a little girl... She would lean into him and she held his arm the entire time."
The alleged couple are believed to have been friends "for a long time", with an insider claiming that there was always the potential for a romantic relationship.
"There's always been this strong chemistry between them," the source said. "This was just a matter of time!"
In late August, Gosling helped put an end to a New York City street fight, stepping into an argument between two men on St Mark's Place in Manhattan's East Village.
The Place Beyond the Pines began production over the summer. It is scheduled for a 2013 release.
Watch Ryan Gosling in the trailer for Drive below: