Hart revealed on Thursday he thought it was unlikely he would play for Manchester City again having been given the cold shoulder by boss Pep Guardiola following his arrival at the Etihad Stadium in the summer.
The signing of Claudio Bravo from Barcelona prompted Hart to move, the England international making the surprise decision to join Italian side Torino on loan for the season.
The Granata faithful have quickly taken the 29-year-old to their hearts and Hart says he has settled quickly in Turin, prompting Petrachi to suggest that a permanent switch might not be out of the question.
"We'll talk about it further down the line, but he's enthusiastic," Petrachi told Radio Sportiva. "He's really happy with the quality of life in Turin and he wants to stay here. It's premature to talk about it now, though, because there are other steps to take first.
"He's a goalkeeper who still has room for improvement and we are counting on him."