Tigers manager Steve Bruce revealed the extent of the injury to Snodgrass ahead of the club's Europa League qualifier against Lokeren in Belgium on Thursday, a club spokesman confirmed.
And it could get worse for Snodgrass who is due to undergo scans to investigate possible ligament damage.
Snodgrass was one of Bruce's big summer signings after his £7million move from Norwich.
The 26-year-old pulled up with a suspected twisted knee in the first half of his side's 1-0 Barclays Premier League win at QPR on Saturday.
Snodgrass wrote on Twitter: "Gutted ....!!! But I will work as hard as I can to come back fitter and stronger."