Earlier this week, the 28-year-old confirmed that he will be leaving the Old Lady on the expiration of his contract, bringing to an end his seven-year spell at the club.
Manchester United and Arsenal are believed to have expressed an interest in Dybala, with Serie A rivals Inter Milan also keen to sign the Argentina international, with a reported four-year deal in place with the Nerazzurri.
However, La Repubblica reports that Jose Mourinho's Roma have overtaken other parties in the race to capture the signature of Dybala, who is believed to have said that he would never choose to join rivals Inter.
Roma can not secure Champions League football next year, but they are likely to be able to offer Dybala a Europa League adventure, and Mourinho is claimed to be keen to get a deal over the line.
Dybala has netted 115 goals and registered 48 assists in 292 appearances for Juve, after arriving from Palermo in 2015.