Aluko has been sidelined since mid-January but had been expected to return to action at the end of this month.
However, after feeling some discomfort on his return to training last week, the 24-year-old has been instructed to rest completely for a fortnight with boss Steve Bruce fearing surgery is the most likely outcome.
"It's not looking good unfortunately, it looks as though he will need an operation," Bruce told Hull's official website.
"It's a huge blow and I felt physically sick on Monday when we found out. That's not just for the team, but for Sone too because he's had a wonderful start to his career here.
"We are going to give him a couple of weeks off and totally switch him off and see if that has any impact. I can't see that being the case, but you never know.
"It's a huge blow, but he'll be back and it's amazing what they can do these days. We'll see over the next few days."
Bruce also revealed defender Paul McShane will be out for "at least a couple of weeks" with a hamstring injury.