Shaw has been absent with an ankle problem since the end of April, while Young also missed the end of last season after picking up an injury in the Europa League semi-finals against Celta Vigo.
The duo travelled to America for United's pre-season tour, however, and the pair made their comebacks for the Under-23 team against Swansea City Under-23s at Leigh Sports Village on Monday night.
Mourinho has said that Young is ahead of Shaw when it comes to fitness, while the boss also insisted that "there is no rush" with Marcos Rojo, who is still on the sidelines with a knee problem.
"Young will come back earlier than Luke. We need our squad, we need everybody, but there is no rush with Rojo. There is no limping. Just straight runs and repetitions. We have to respect the timings and they are bringing him to the team around December," Mourinho told MUTV.
United will look to make it three straight wins at the start of the new Premier League campaign when they welcome Leicester City to Old Trafford on Saturday night.