A source said the new deal from the Bears was below the market price for an NFL quarterback with Grossman's experience and came after discussions between the his agent and Bears general manager Jerry Angelo, The Chicago Tribune reported Saturday.
A league source told the Tribune the offer is not expected to surpass $2 million even with incentives.
Friday's deal came one day after Bears head coach Lovie Smith said he was confident Grossman's best days on the field were ahead of him.
"There have been highs and lows and we think Rex is a good football player and I choose to remember those highs," Smith said Thursday at the NFL scouting combine in Indianapolis. "I still don't think he has peaked and still think his future is bright."
Both Smith and Angelo have remained steadfast in their stance that if Grossman does remain with the Bears, he would have to compete with quarterback Kyle Orton for a starting position, the Tribune said.