The 28-year-old forward's Juve contract expired this summer and he had been touted for a move to Serie A runners-up Inter Milan, which would have ensured Champions League football next season.
Instead, the Argentine will join Europa Conference League winners Roma, who finished sixth in Serie A last season, Gazzetta Dello Sport and Sky Sport Italia reported.
Dybala has been pictured in Portugal with Roma's sporting director Tiago Pinto and will undergo a medical and sign a three-year contract worth six million euros ($6.08m) per season including bonuses, Sky reported.
Dybala spent a trophy-laden seven years in Turin but has been dogged by injuries over the last two seasons.
He joined Juve from Palermo in 2015 going on to win five Serie A titles and four Italian Cups.
He also played in the team which reached the 2017 Champions League final lost to Real Madrid.