Cutler suffered a concussion on the apparent head shot delivered by Dobbins in Sunday's 13-6 Houston victory over the Bears at Soldier Field.
The Bears signal-caller didn't return after halftime and his status for Monday's game against San Francisco was not clear.
ESPN and NFL.com cited league sources they did not name as saying Dobbins had been assessed the hefty fine, which he is likely to appeal.
Dobbins told reporters after the game he felt it was a clean hit to Cutler's chest, but replays appeared to show Dobbins' helmet striking the quarterback's chin, constituting banned head-to-head contact.
Bears Coach Lovie Smith said Wednesday Cutler was "feeling a whole lot better," but didn't say whether he would start next week, the Chicago Tribune reported.
The newspaper said Cutler was at the Bears' complex Wednesday the team's staff was "still evaluating him" -- and noted the team isn't resuming practice until Thursday.
"He's meeting with people, seeing medical staff, our trainers and all of that. But he is getting better. We do have a little bit more time and that is always good."