The deal would reportedly protect the Tigers from losing Brady to other college assistant jobs. The 30-year-old is highly regarded after helping mastermind quarterback Joe Burrow's historic season.
But the extension gives Brady flexibility to leave LSU for a college head coaching position or a job in the NFL. The Carolina Panthers are reportedly eyeing him for their offensive coordinator position.
Brady's previous salary of $410,000 is expected to be doubled in his new deal.
The former New Orleans Saints assistant joined LSU during the offseason to implement the spread offense, which has been an instant success.
LSU is set to play in the national championship game against Clemson after leading the nation in points per game and yards per game this season.