The Catalans were apparently looking to get him off the books, with head coach Xavi also using him sparingly this season, preferring the 19-year-old Alejandro Balde over him.
Alba has made three appearances in La Liga so far, starting just once and seeing a total of 126 minutes of action as his future at the club is up in the air.
Lionel Messi was aware of Barcelona's attempts to offload the player and reportedly wanted the Parisians to swoop in for the Spaniard.
However, El Nacional further reported that Al-Khelaifi feels Alba's performance levels over the last few years haven't been impressive while the left-back is edging closer to retirement.
Messi, although disappointed at his club vetoing a move for his former teammate, accepted their decision.
Even if a move to PSG materialized, Alba's chances of breaking into the first team would've been slim as head coach Christophe Galtier trusts Nuno Mendes at left-back.
He also has Juan Bernat as an option in that position, while Achraf Hakimi and Presnel Kimpembe are also capable of operating from the left-back position.
Alba has another two years on his contract with Barcelona, but his time as a first-teamer appears to be over, with new signing Marcos Alonso also seemingly ahead of him in the pecking order.
Lionel Messi in fine form for PSG this term
Following a difficult first year at the club, Lionel Messi seems to have fully acclimatized to life in Paris as he's started the new season with a bang.
With four goals and seven assists from just nine games, the Argentine ace has nearly matched half of his goal-contribution from last season already.
He's continued his role as the creator-in-chief of the side, setting up the likes of Kylian Mbappe and Neymar with incredible assists.
While no player in Ligue 1 boasts more assists than Messi right now, his partnership with the aforesaid attacking duo has also improved. It's still early days but so far so good for the seven-time Ballon d'Or winner.