The Wales international, who has scored 22 goals this season, was injured during the 2-0 defeat to Sky Bet Championship leaders Leicester City at Turf Moor on Saturday.
Clarets boss Sean Dyche was hopeful the injury, which came after an innocuous challenge early in the first half, was not that serious, but scans have revealed the full extent of the damage and Vokes will now see a specialist.
Dyche told his club's official website: "Sam has had a great season and I am sure you will join us all in supporting him during his recovery.
"As ever, our team mentality remains focused and this represents another opportunity for someone to come and make a continued contribution in the closing weeks of the season."
Burnley's leading goalscorer Danny Ings is also on the sidelines at present but is making good progress in his recovery from an ankle injury and is expected to rejoin the squad soon as they continue their push for promotion to the Premier League.