Iranian media had earlier reported that Cheshmi has reached an agreement with Qatari giant Al Ahli.
Now, Basaksehir has reportedly set its sight on signing the 25-year-old player.
Cheshmi was a member of Iran national football team in the 2018 World Cup and the 2019 AFC Asian Cup.
Cheshmi is facing a dilemma over his transfer to Qatar Stars League or Süper Lig.