The Turkey international is widely expected to leave Camp Nou this month having failed to make a single appearance for the Spanish giants during the 2017-18 campaign.
Barca's acquisition of Philippe Coutinho from Liverpool has only served to increase speculation linking Turan to a departure from the La Liga outfit.
Basaksehir president Goksel Gumusdag has claimed that Turan wants to join the club, who are top of the Turkish Super Lig, with discussions over a deal reportedly underway.
"I spoke to Arda Turan face to face in Barcelona," Gumusdag wrote on Basaksehir's website. "He wants to come to Basaksehir. I saw his passion and he told me his goals. After that, we met with Barcelona CEO Oscar Grau.
"Barcelona's approach is positive about this transfer. We would like to sign Arda, despite the high cost. Our two sponsors are willing to help with the transfer process.
"I understand the media are excited and I'm so excited too, but we have to wait. We have a couple of steps to go in order to finish the deal."
Turan has also been linked with Arsenal, Everton, Besiktas, Galatasaray, Besiktas, AC Milan and Chicago Fire.