Financial details were not disclosed, but ESPN reported that it is a five-year, $42 million offer that includes $26 million in guaranteed money.
Mack would reportedly get $10 million this year and $8 million in each of the final four seasons of the deal. According to other reports, there is also a stipulation that the team cannot place the franchise tag on him after the 2015 season and a no-trade that runs through the first three years of the contract.
The Browns' decision was not a surprise. Owner Jimmy Haslam said earlier the team would match any offer "in a second" and that Mack would be a core element in a rebuilt offense.
"We have talked about keeping our own players and this is a positive for us," Browns general manager Ray Farmer said in a statement. "Alex is a quality person and player that truly brings to life what playing like a Brown means.
"The ending is positive for everyone. Keeping our young, good nucleus of players is vital for clubs and specifically the Browns, and therefore is this a good step."
The Browns placed the transition tag on Mack -- not the franchise tag -- giving the team the chance to match any offer sheet he signs. The transition tag guarantees Mack the average of the 10 highest-paid offensive linemen, or $10.039 million.
The 6-foot-4, 311-pound Mack was drafted by the Browns in the first round in 2009. He has not missed one offensive play in five years, a run of 4,998 consecutive snaps.
The Jaguars were looking at Mack to replace Brad Meester, who retired after 14 seasons with the Jaguars.