Alba, 32, has a contract with Barca until 2024 and a €500 million buyout clause, but he is one of the club's biggest earners at a time when they need to reduce their wage bill.
Italian champions Inter are one of the teams to have shown an interest in signing him, as first reported by Mundo Deportivo and confirmed by ESPN.
While a source close to Alba said he is not even contemplating a move this summer, Barca's need to trim outgoings means they would consider offers for much of their squad.
La Liga president Javier Tebas has warned Barca they must decrease spending if they want to register Lionel Messi's new contract -- with the club optimistic an extension will be agreed soon -- and new signings Sergio Aguero, Eric Garcia, Emerson Royal and Memphis Depay.
There are plans to meet with some of the squad's more experienced players to discuss further salary cuts. Gerard Pique has said in public he would have no problems dropping his pay, but Alba, Sergi Roberto and Sergio Busquets have not yet commented on the issue.
Both clubs are struggling financially, though, so any potential deal would have to be a player exchange and sources told ESPN that Barca like centre-back Bastoni.
Barca have already signed Eric Garcia to strengthen their defence this summer, but ESPN previously reported that they would like to add another defender if possible. Manchester City's Aymeric Laporte is also a target if they can first move on Clement Lenglet and Samuel Umtiti.
Relations between Barca and Inter are good, according to sources. The two clubs held talks over the transfer of Lautaro Martinez last summer, a deal which never came to fruition, while midfielder Vidal swapped Spain for Italy.
Meanwhile, regardless of what happens with Alba, Barca remain in the market for a left-back, with Junior Firpo one of the players earmarked to leave the club this summer. Umtiti, Philippe Coutinho, Martin Braithwaite, Neto, Matheus Fernandes and Miralem Pjanic are also considered surplus to requirements.