The 29-year-old threw for 3,385 yards with 14 touchdowns on his debut season in the NFL, in which he started 15 games for the Browns.
When asked at a press conference if Weeden would remain the starter, head coach Rob Chudzinski said: "I think it's premature."
Offensive coordinator Norv Turner added: "A lot of [quarterbacks] have had their best seasons when I was coaching them. He has a lot of the skill set we're looking for. Certainly has a big arm."
The Browns finished the 2012 season last in the AFC North, with five wins and 11 losses.