16-year-old Scarlett assisted Carlos Vinicius for the final goal of Tottenham's 4-0 victory over Wolfsberger on Wednesday.
Mourinho worked alongside Rashford at Old Trafford and sees similar traits in the promising Scarlett.
He said on BT Sport: "He is a diamond, a kid with incredible potential. He has worked many times with the first team and that gives him a different personality.
"He is still 16, 17 soon and I believe next season he will be a first team squad player because he has a lot of talent.
"He is going to be a fantastic player and I hope everything around him goes well.
"He is a striker, a number nine, I have been playing him from the sides similar to Marcus Rashford and is very clever."