Juve said that Weah underwent tests on Wednesday morning "which revealed a low-grade injury of the femoral biceps of the right thigh".
Weah, the son of former Ballon d'Or winner and ex-president of Liberia George Weah, suffered the injury during his nine minutes as a substitute in Juve's 4-0 thumping of Cagliari in the Italian Cup on Tuesday.
He is expected to miss Juve's Serie A matches with Monza and Fiorentina before Thiago Motta's team travel to contest the Italian Super Cup in Saudi Arabia, where they face AC Milan in the semi-finals on January 3.
The 24-year-old, who has 42 international caps, has played 18 times for Juve in all competitions this season, scoring four goals and setting up four more.
Juventus are sixth in Italy's top flight, nine points behind current league leaders Atalanta.
