United are currently skippered by Wayne Rooney, with veterans Michael Carrick and Zlatan Ibrahimovic in reserve, but the trio are unlikely to be considered first-team regulars at Old Trafford in the long term.
That will leave Mourinho, or another manager, with a decision to make regarding the club's next captain, and the Portuguese has said that Pogba would be in contention for the position after impressing in his opening five months in the North-West.
The 53-year-old is quoted by The Mirror as saying: "I think he has. I think he has the charisma, the ambition, the mentality. He's a good professional. Being so young and with so many ingredients, I think he can."
Pogba has scored six times from 27 appearances in all competitions since his world-record move from Juventus in the summer.