Totti has been in contact with Roma for some weeks about returning in a management role.
Gasperini dined with Totti last weekend, with former teammate and current Roma ambassador Vincent Candela also present.
Talks on Sunday evening involved Gasperini making clear to the Roma legend how much he is wanted back at the club - and what his job would involve.
La Gazzetta dello Sport says Totti's responsibilities would include being involved in technical decisions, scouting for talent across Europe, and even convincing potential transfer targets.
Gasperini said last month: "For me, he would be a huge asset, a significant opportunity: I have several ideas about what role to give him and how he could be developed."
