The Spanish left-back has impressed at the Nou Camp in recent seasons and the Special One has noticed.
But Don Balon say Alba has no intention of joining Mourinho at Old Trafford because of his partnership with Lionel Messi.
United have several options at left-back but nobody has grabbed their chance to make the position their own.
Luke Shaw, Daley Blind, Matteo Darmian and Ashley Young have all been used by Mourinho at different points during his tenure.
Alba is the player United are desperate to bring in but the reports say the 28-year-old has turned down a mega proposal from Old Trafford.
Messi is said to have an excellent relationship with Alba, a key reason as to why the defender has no desire to leave the Nou Camp.
Don Balon add that the Spain international has made it clear to Mourinho no offer will make him leave Barca for United.
Alba has made six La Liga appearances so far this season, providing four assists.
Barcelona sit top of the table after winning all seven of their matches so far.
Ernesto Valverde’s side are already seven points clear of last season’s champions and arch rivals Real Madrid.
United return to Premier League action after the international break against Liverpool at Anfield.
The Red Devils currently sit second in the table on goal difference behind Manchester City.
