Ahmet Bulut claims that Barca are in talks with Turan over a move to the Nou Camp amid interest from the likes of Monaco and United.
United have been heavily linked with the Turkish international this summer, but national newspaper Milliyet claims he is close to finalising a move to Barca. Arsenal and Chelsea have also been mentioned as potential destinations for the attacking midfielder.
The Catalan giants are willing to meet Turan's massive release clause, and personal terms are likely to be a formality.
It would be another massive blow to Atletico Madrid, who have been left helpless as several of their top stars look set to move on. Diego Costa is expected to join Chelsea, and he could be followed by Felipe Luis and Tiago.
Thibaut Courtois could return to Stamford Bridge for next season and Koke has been linked with United among others.
Turan, 27, was a key part of Diego Simeone's La Liga winning team last season and he has 72 caps for his country.
Barcelona are undergoing a transition this summer with a new coach in the form of Luis Enrique. They will also be without Carles Puyol for the first time since 1999 as the Spain and Barca legend has retired.
Javier Mascherano recently signed a new contract, and the club is expected to be active in the transfer market this summer following a season during which they failed to win a major trophy for the first time since 2008.