Bulger took a hit Sunday from Aaron Kampman in a 36-17 loss to Green Bay. He left the game late in the first quarter and did not return.
The Rams next play San Francisco.
"We'll see where he's at tomorrow and Wednesday and if he'll do anything at all but that's what it is," Spagnuolo said. "It's not torn; it's a bruised rotator cuff. I don't want to speak for him but I know he's very sore today and we're going to be careful with it. If he can throw Wednesday, he will, if he can't he won't."
Bulger has thrown for 339 yards and a touchdown on 35-of-68 passing in three games this season.
Kyle Boller took over in Bulger's absence Sunday and completed 16-of-31 passes for 164 yards and two touchdowns.