Savarese yesterday confirmed to Tribalfootball.com that contact had been made with Totti about a move to the Big Apple.
The Cosmos coach is aware of competition from Leicester for the Roma captain's signature and told Tribalfootball.com: "I think that definitely the name of the Cosmos worldwide has big relevance and that shows on a constant basis the work we have done in the few years since we have brought the club back.
"I've been able to be respected and the fact we were able to bring Raul shows what kind of organization we are and want to be.
"It's good to see that we can be put out there and be taken seriously in situations like this."
Cosmos are considering Totti, who is off contract in June, as a replacement for Real Madrid icon Raul, who hung up the boots after leading the club to the NASL title last season.