United are known to be in the market for a left-sided defender during the current transfer window, with manager Jose Mourinho still unconvinced by his options in that position after preferring makeshift full-back Ashley Young to the likes of Luke Shaw, Matteo Darmian and Daley Blind for much of the 2017-18 season.
Chelsea, meanwhile, could consider a move for the 29-year-old as an upgrade on current wing-back Marcos Alonso, despite the Spaniard having impressed since his arrival at Stamford Bridge.
Spanish publication Sport reports that Alba could be tempted by a move to the Premier League, particularly as he has grown increasingly frustrated with protracted transfer talks at Camp Nou.
The player was one of Barcelona's star performers during their double-winning season, registering 11 assists for the club, eight of which were for Lionel Messi.
Alba's preferred choice is to remain with Barcelona, but United and Chelsea are thought to be monitoring his situation ahead of a potential summer deal.