The Sun report that Arsenal are leading the race for the Turkish international and that Barca want £20 million.
Arsenal boss Arsene Wenger reportedly wants to deploy Turan alongside Mesut Ozil and Alexis Sanchez in an attacking midfield trident.
Turan joined Barcelona on a five-year deal from Atletico Madrid last season but had to sit the first six-months out on the sidelines.
Despite missing the first half of the season Turan still made 25 appearances in all competitions and lifted the league title as well as the Spanish Cup.
Turan has been recalled to training earlier than expected as Barcelona look to get the Turkish international back into shape ahead of the upcoming season.
There are conflicting rumours from Spain regarding whether Barca have transfer listed Turan however, his agent Ahmet Bulut told us that his client is happy at the club.
Bulut did however, state in a more recent interview that he could not rule out the possibility of Turan leading over the summer – even though he is happy at the club.
The agent told Calcio Mercato: “I know nothing of Milan’s interest in Arda. At the moment I can only say that he wants to remain with Barcelona. I don’t know, however, what could happen during the transfer window.”