Palmer spent the pre-season with the NFC North side and has been brought in as cover for starter Jay Cutler, who will miss the next month with a groin injury, while Grant will add depth in Lance Briggs's absence through shoulder injury.
Head coach Marc Trestman told the team's official website: "[Palmer] became very affluent in our offensive language very quickly.
"He was very motivated on his own to get himself ready because we were getting ready for a season at the time we were getting ready to play the last pre-season game.
"He impressed us. He was very efficient running the offense when he was in there on very short notice. We feel good that he's available and we can bring him in here to back up Josh [McCown]."
The Bears play the Green Bay Packers next Monday.