The Chile international was taken off on a stretcher in the Latics' 3-2 win over West Bromwich Albion on Saturday.
Roberto Martinez has revealed that the injury "doesn't look great" and said that he expects the 28-year-old to be out for 10 days - meaning that he is likely to miss the Premier League clashes against Swansea City and Arsenal, as well as the final.
"Jean is the negative," said Martinez after the game. "He pulled his hamstring - a soft tissue problem in the left leg.
"We'll have to assess him but he doesn't look great. This sort of injury is normally 10 days but we'll see if we can get some magic hands on him."
Beausejour moved to the DW Stadium from Birmingham City in 2012.