Cavani's contract at Manchester United will officially expire at the end of June, while Di Maria's deal with Paris Saint-Germain is also heading into its final week.
Barcelona are pushing to sign Bayern Munich striker Robert Lewandowski during this summer's transfer window, but according to Fichajes, if they are unsuccessful in bringing the Poland international to Camp Nou, the club could make a move for Cavani.
The Uruguay international is said to be desperate to play in La Liga next season, and Barcelona president Joan Laporta is believed to be open to bringing him to Camp Nou for the 2022-23 campaign.
The 35-year-old struggled with injury problems during the 2021-22 season, managing just two goals and one assist in 20 appearances for the Red Devils in all competitions.
Cavani scored 19 goals and registered seven assists in 59 appearances for Man United in total, though, having moved to Old Trafford on a free transfer in October 2020.
Barcelona could lose Ousmane Dembele on a free transfer this summer, while Martin Braithwaite and Memphis Depay might both depart, which would weaken the club's attacking options for next term.
Cavani is believed to be desperate to continue playing at the highest level next season rather than look for a move to the MLS, as he will represent Uruguay at the 2022 World Cup and wants to be in the best possible shape to help his country in the competition.
Meanwhile, according to Sport, Di Maria has emerged as an option for Barcelona, with the Argentine set to become available when his PSG deal expires at the end of the month.
The report claims that Juventus are currently leading the race for the South American, but the player is holding out for a move to Camp Nou ahead of the 2022-23 campaign.
Di Maria is said to be viewed by Xavi's side as a low-cost alternative to Raphinha, with Leeds United placing a high valuation on the Brazil international.
The 34-year-old previously played in Spain for Real Madrid between 2010 and 2014, representing the club on 194 occasions in all competitions before making the move to Manchester United.
Di Maria has been with PSG since 2015, and he was again a regular for the club during the 2021-22 campaign, making 31 appearances in all competitions, contributing five goals and nine assists in the process.
The attacker won one La Liga title, two Copa del Rey crowns and the Champions League during his time at Bernabeu, and a return to Spain is said to appeal to him this summer.
Juventus are believed to be waiting on a final decision from Di Maria, having already offered him a one-year contract with the option of a further 12 months.