Broncos coach Mike Shanahan said both are expected to be placed on injured reserve. The team has lost seven tailbacks this season.
Denver is expected to bolster its depth at tailback by activating Cory Boyd off the practice squad and signing Alex Haynes, who was waived by the Broncos on Nov. 22.
Pope became the sixth tailback to start in the Broncos' battered backfield Sunday against Buffalo but went to the sideline in the first quarter with a torn left hamstring. He finished with six carries for 44 yards.
He's expected to be out six to eight weeks.
Young picked up the slack for Pope before leaving with a pinched nerve in his neck late in the third quarter.
An MRI on Monday revealed Young had a ruptured disk
"Hopefully the disk will go back into position where he won't have an operation," Shanahan said.
The injuries to Pope and Young left Tatum Bell as the only healthy running back. Bell carried the ball five times for 20 yards.