The deal is understood to be for five years and could be worth up to $42m (£25m), with $26m (£15.5m) guaranteed.
Mack would make $10m (£6m) in his first season with the team before making $8m (£4.8m) in each of the next four seasons, with his guaranteed amount of money the most offered to any free agent this off-season.
The Browns now have five days to match the offer from Jacksonville, and they are expected to do so as they look to keep hold of the two-time Pro Bowler.
The contract offered to Mack by the Jaguars stipulates that he can void the final three years with the promise that he will not be given the team's franchise or transition tag.
The 28-year-old has been with the Browns for five seasons having been picked in the first round of the 2009 NFL Draft.