Conte was sacked as Tottenham Hotspur boss last month following a shocking outburst in his press conference after his side drew 3-3 at Southampton.
The Italian boss has had great success domestically in his homeland, winning four Serie A titles as a manager with Juventus and Inter Milan.
Now, CalcioMercato claims that the board at Roma have identified Conte as the man to come in should their current Portuguese boss leave.
Mourinho has been rumoured to be on the way out at Roma, after some disputes between himself and the club, and the possible scenario where Paris Saint-Germain may come in for him in the summer.
After giving Mourinho substantial backing in the transfer market at the start of last season, spending more than any other club in Serie A to bring in the likes of Tammy Abraham, Rui Patricio and Marash Kumbulla, club chiefs at Roma are said to be willing to give Conte similar funds to spend in the market.
Roma have not won the Scudetto since 2001, so Conte's recent success at achieving this feat reportedly makes him an appealing candidate for the Giallorossi.