But Mendes, who has openly said she doesn't believe in the institution, may prove difficult to persuade.
"Ryan is head over heels in love with Eva, and he wants to have a life with her, including children," a source told the National Enquirer. "It's something Eva has balked at in the past, but Ryan has a charming way of helping her see a different kind of future."
Gosling, 31, and Mendes, 38, met on the set of their forthcoming film "The Place Beyond the Pines," a crime movie about sin and morality that was written and directed by Derek Cianfrance. Cianfrance also directed Gosling in "Blue Valentine," which resulted in a breakout performance that earned the star an Oscar nod as best actor.
The drama, split into three different movements that span 15 years, premiered at the Toronto Film Festival to mostly positive reviews by critics. Tim Robey, in a review for the Telegraph, wrote that no other film at the festival "started more punchily or sustained such swaggering visual allure." He called it "dense and enthralling."
Mendes plays Gosling's on-screen girlfriend in "The Place Beyond the Pines," and she said she had "never felt so creatively satisfied" before with a co-star. But whether their chemistry will be enough to change the actress' stance on getting married is another matter.
According to the National Enquirer's source, Mendes is "getting on board with the idea of marriage." In May, however, the actress appeared to sing quite a different tune when she discussed her marriage views with Chelsea Handler on the television talk show "Chelsea Lately."
"Last time you were here, we had an interesting conversation about relationships, because you and I kinda have similar views on marriage ... and avoiding it," joked Handler.
Mendes laughed and replied: "The thing is, I'm not like anti-marriage or the institution, like for other people. And I like the idea of a union, that's all very beautiful. But I just think you know, it's a very old-fashioned, archaic kind of thing. I mean, we did it for land, originally -- how unromantic is that?"
Handler went on to share an anecdote about how she felt the same way in a past relationship, before realizing she simply had no desire to marry that specific boyfriend. "So it could depend," said Handler.
"No, not for me," responded Mendes. "It's just the institution. ... If you love someone and you have a spiritual union, I just don't think a piece of paper, because society tells us ... it's just so not about what it should be about."
According to the National Enquirer's source, Gosling has been changing Mendes' views by focusing on the idea of raising a family together. "He is now confident when he pops the question she will say, 'Yes,'" said the source. "Ryan wants children, as does Eva, but also believes in marriage and youngsters growing up knowing mommy and daddy are committed to each other."
And Gosling isn't the only one who's in love. In May, In Touch magazine reported that Mendes' mother "gushed" after meeting Gosling. "As she was leaving, Eva's mum was hugging and kissing Ryan," reported the magazine. "She loves him!"