The Grizzlies said Allen was hurt when he fell awkwardly on his left leg, which forced him to leave Friday's win over the Detroit Pistons.
In his second season, Allen is averaging 7.4 points, 2.2 rebounds and 1.4 assists.
The Grizzlies also announced they have recalled Josh Jackson from the G League.
Jackson, the fourth overall pick in the 2017 draft, was traded by the Phoenix Suns to the Grizzlies this offseason after two uneven seasons in Phoenix.
Jackson, however, has yet to play for the Grizzlies as he has spent the entire season playing for the Memphis Hustle.
The 22-year-old has averaged 20.3 points, 7.5 rebounds and 4.3 assists in 26 games for the Hustle, and was suspended one G League game by the organization in December for a violation of team rules.