Per the source, Barca are approaching their budget after splashing out for Umtitti, Digne and Denis Suarez. The La Liga champions may also sign Andre Gomes from Valencia.
Barca will have to raise funds for any additional transfers and as a result will consider offers for Turan.
The Turkish international midfielder joined on a five-year deal from Atletico Madrid last summer on a deal worth £33 million.
The report claims that Barca would accept two thirds of the price they paid for him last season.
Per the source, Arsenal star Alexis Sanchez is also reported to have got involved and urged his side to sign the Turkish international.
Despite the constant rumours that Turan will leave Barca over the summer his agent Ahmet Bulut says his client is happy at the club.
Turan is currently training with the rest of the squad and is expected to travel to Ireland for Barca’s pre-season friendly with Celtic in Dublin next week.