The daily Turkish sports paper states that Barcelona are willing to listen to offers in the region of €25 million.
Fanatik goes onto state that the two London based clubs are ready to table bids for Turan.
The Daily Star also report that Premier League giants Arsenal and Chelsea have both been linked with a move the the 29-year-old midfielder.
Meanwhile according to the Spanish outlet Sport, 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.
Turan has another four-years remaining on his contract with Barcelona. The 29-year-old provided four assists and scored two goals in 25 appearances in all competitions for Barcelona last season.