The £7million summer signing from Norwich picked up the injury on his league debut against Queens Park Rangers on Saturday.
Should a six-month lay-off be confirmed, Snodgrass would also miss Scotland's four opening European Championship qualifiers, as well as a friendly against England in November.
Hull manager Steve Bruce said: "He dislocated his kneecap and damaged the ligaments which hold it in place.
"It looks as if it will be a repair job and an operation.
"We'll know more later today because he's seeing a specialist in London. The odds are he'll need an operation and that will be six months."
He went on: "We didn't envisage losing Snodgrass for six months, which is a huge blow to us.
"He was brought in to give us that little bit of quality in the midfield area."
Snodgrass wrote on Twitter: "Gutted ....!!! But I will work as hard as I can to come back fitter and stronger."