Despite not practicing during the week, Murray is listed as doubtful for Sunday's game against Cleveland. He has missed the last four games with the foot injury.
Murray was able to do more on-field rehab work this week but has not practiced since suffering the injury Oct. 14 at Baltimore.
He said earlier in the week this is the best his foot has felt since he hurt it, but the Cowboys want to make sure he is healthy enough so he doesn't suffer a setback once he returns to game action.
Coach Jason Garrett said Murray would not need a lot of practice to play, but with a quick turnaround before the Thanksgiving game against Washington, it is unlikely he would play against the Browns.
Felix Jones would start for the fifth straight week if Murray is out. Jones is coming off a 71-yard rushing effort against Philadelphia, his highest total since taking over for Murray.