The Brazil international has not been offered a new contract with Barca, with his current deal expiring at the end of the season.
Spanish radio station Radio Marca claims the 31-year-old has signed a three-year preliminary agreement with the French champions.
Alves, who is also a reported target of several Premier League sides, has made 32 appearances this season for the current leaders of La Liga, including 19 league starts.
The South American, who joined the Catalan outfit in 2008 from Sevilla, has won four Primera Division titles, two Copa del Rey trophies and the Champions League twice.
According to Catalan Sports newspaper El Mundo Deportivo, Barca have set their sights on Villarreal right-back Mario Gaspar as a possible replacement for Alves.
The 24-year-old, who is tied to Villarreal until June 2019, has enjoyed an impressive campaign with the Valencian outfit.
Gaspar has scored three goals and set up one more in 22 La Liga appearances to help Villarreal climb to sixth in the standings.
Barca cannot make any signings until 2016 after being handed a transfer ban for breaching FIFA’s rules on the transfer of players aged under 18.