The Pittsburgh Steelers QB has won two championships since joining the AFC North side in 2004, while Bradshaw has four Super Bowl wins.
Speaking to reporters in London while promoting the NFL International Series, Roethlisberger said: "I want to pass Terry Bradshaw by getting more Super Bowl wins.
"That's our expectations in Pittsburgh. There is no other franchise I would rather play for than the Pittsburgh Steelers."
Roethlisberger has a contract with the Steelers which will keep him at the team until the end of the 2015 season.