Azpilicueta, 32, has made 12 appearances for Chelsea this season and remains a regular under coach Thomas Tuchel after captaining the side to Champions League success last season.
However, with his deal up next summer, the Spain international is yet to commit to new terms at Stamford Bridge, which has alerted Barca, who have shown an interest in the defender in the past, too.
Sources have told ESPN that Chelsea remain in negotiations with Azpilicueta and are hopeful he will extend but there is not yet an agreement for him to prolong his stay in London.
If Azpilicueta chooses to run down his contract with Chelsea, though, new Barcelona coach Xavi Hernandez would be especially keen on securing him.
Sources have told ESPN that Xavi values his ability to play at right-back in a four-man defence or on the right of a back three. He has even played as a centre-back in a back four at times for Spain.
Alves, 38, has joined on a contract until the end of the season initially after leaving Sao Paulo in September.
Azpilicueta, who previously played for Osasuna and Marseille, has been at Chelsea since 2012. He has made over 400 appearances for the Premier League side and has recently returned to the Spain setup under Luis Enrique.
He has made 36 appearances for his country, with 11 of them coming in 2021 following a three-year absence from international football.