Despite Cowboys coach Jason Garrett confirming the procedure will take place, Murray remains a chance to take his place in the side against the Indianapolis Colts on Sunday.
"The biggest question we have to ask ourselves is, is he functional to do his job?" Garrett said.
"Can he hold the football? Can he carry it under duress? Can he block? Can he do the things necessary to play the position?
"He's as strong-willed and as determined an individual as I've ever been around, and if anybody has a chance to come back, he does.
"So let's take first things first and see how the surgery goes, and we'll make our best decisions from there."
Murray leads the league in gains rushing with 1,687 yards. Joseph Randle would appear to be the most likely candidate to replace Murray over Lance Dunbar.
Randle has had limited opportunities - no one has with Murray dominating the league with 351 carries - but when given a chance, Randle has averaged seven yards a carry this season, including 6.5 in the 35-27 win over the Philadelphia Eagles on Sunday.
The timing could not be worse for Dallas, which moved into the NFC East lead after edging the Eagles.
They have the Colts and Washington Redskins remaining in the regular season.
The key to the Cowboys success this year can be traced right back to leaning on Murray and not putting the entire offense on Tony Romo's shoulders.
Murray broke Jim Brown's record set in 1958 with seven straight 100-yard games to open the season.