As Puck reports, McConaughey auditioned alongside Kate Winslet, according to film producer Jon Landau in his book. The filmmakers wanted to test whether they'd have on-screen "chemistry." The actress was happy with her potential co-star, but James Cameron wasn't impressed.
The director didn't like the actor's Southern accent. But when Cameron asked Matthew to run the scene again differently, he confidently said "no," explaining that he'd already done it well enough. The filmmaker immediately passed on him.

Source:
imdb