The 29-year-old told The AP: "I want to compete, and having Jason here is better for everyone involved because we'll push each other.
"This is probably the biggest year of my career because I've got to show that I can play at this level and make adjustments.
"Now that it's my second year here, it's my time to take control and be the guy. They're not going to stick me out there if they don't think I'm the guy who can win games."
Weeden started 15 games for the Browns in 2012, recording 3,385 passing yards.