Citing sources, ESPN said Alba feels settled at Barcelona following the appointment of Ernesto Valverde as coach after being overlooked for key games by previous coach Luis Enrique last season.
The 28-year-old would only join United if Barca decided he was surplus to requirements, the report added.
Red Devils boss Jose Mourinho has already brought in defensive reinforcements in the form of centre-back Victor Lindelof, signed from Benfica for £31m ($40m) earlier this month.
But with Luke Shaw and Marcos Rojo currently injured – with ankle and knee problems, respectively – the Portuguese coach must decide whether to invest in a new left-back.
Matteo Darmian, Ashley Young and Daley Blind were employed at left-back at various points last season, but none of the three are naturals in the role.
Alba said in an interview in March that he was frustrated at being overlooked by Enrique for some crucial Champions League knockout games in which the Spanish coach opted to play with a three-man defence.
"All players want to have continuity, to have minutes," he was quoted as saying by the Daily Mail. "It's true that I've been more involved in the last few seasons [than now], but those are the decisions of the coach. When I played, I did not ask for explanations, and I won't do that now, either.
"I have to work and give everything and when I'm given minutes I have to take advantage of them, as I have always done.
"I think I can play with three defenders. I'm capable of doing it. Obviously my strength is going up and down the wing – that's what has given me the opportunity to play for the national team and at Barcelona."