The sports newspaper reports that Barcelona boss Luis Enrique has requested the club target the Turkish international winger.
Turan is expected to leave Atletico Madrid after his agent Ahmet Bulut confirmed that he is in talks with three Premier League clubs.
“I’m in discussion with three clubs, but I can’t reveal their names, they are three of the best clubs in England and indeed in Europe,” he told AS.
Bulut also stated that Turan will consider his options and make a decision on his future by 10 July.
The 28-year-old winger has also been closely linked with United who are reported to have put forward an offer according to Gazzetta dello Sport.
Turan became the first Turkish international to win the La Liga with Atleti last season and he has also won the Spanish Cup, Europa League, European Super Cup and was in the Rojiblancos side that reached the Champions League final.