The tactician is currently in the middle of a legal battle with his former club Chelsea, claiming he is owed up to €10m, which the Premier League club have no intention of paying.
Conte has long been linked with a return to Serie A, as the ex-Juventus boss is known to be on the radar of both Inter and Milan.
Now Il Tempo and Repubblica newspapers claimed Roma have made the first formal contact with Conte to discuss potential terms.
The Giallorossi fired Eusebio Di Francesco last month and brought in Claudio Ranieri are caretaker for the rest of the campaign.