Dybala picked up the injury while scoring the winner for AS Roma from the penalty spot in their 2-1 success which took them up to fifth on the Serie A standings.
Dybala was taken off for Nemanja Matic in the 50th minute shortly after restoring Roma's lead.
Asked about Dybala's condition after the win, Roma manager Jose Mourinho told DAZN: "I say bad, but I think very, very bad.
"I am not a doctor, but in my experience, and after speaking to Paulo, it's unlikely we'll see him this year."
Dybala has tallied five goals and two assists in eight Serie A matches for Roma since joining on a free transfer from Juventus this summer.
Despite Lecce finishing the game with ten men, Mourinho was unimpressed with the nature of Roma's win.
"There was fatigue, both physical and mental. Playing Thursdays and Sundays is tough," Mourinho said.
"When I see Udinese and Atalanta [who drew 2-2 earlier on Sunday], I understand that they don't play midweek.
"We entered very well, with intensity and ambition. After that it was more difficult because we managed it badly; when you are tired, instead of playing simply you complicate things.
"Today we didn't play well and we won, with Atalanta we played better and we lost [1-0 last month]. We have to manage tiredness better."
