The 48-year-old actor admitted he was desperate for the lead role of Jack Dawson in the movie, which eventually went to Leonardo DiCaprio, and was devastated when he never heard back from producers after his audition.
Speaking on The Hollywood Reporter's 'Awards Chatter' podcast, Matthew said: "I went and auditioned for that. I wanted that. I auditioned with Kate Winslet. Had a good audition. Walked away from there pretty confident that I had it. I didn't get it. I never got offered that."
And Matthew also insisted he doesn't regret his years as a romantic-comedy leading man, revealing he loved going to work at the time.
He said: "It was fun, it was great pay and I learned that, you know what, these are not the type of movies to dig deep [within one's own experience]. They were 'Saturday characters.' I remember saying, 'I'm gonna give myself another Saturday!'
"They were fun. The pay was great. I was looking forward to them. I enjoyed going to work. I was also living on a beach and going out without my shirt on, just like I did before I was famous. I was living a romantic-comedy."
And, although they were fun movies to make, Matthew worked just as hard on the romcoms as he does now on more serious movies.
He said: "I had just as many scribbles and dibbles and options and things written on my scripts and pages tagged and feathered and bent then, in those rom-coms, as I do now. It's just a different kind of work and a different place to dig in."