The veteran forward has been withdrawn from Ireland's final warm-up game against Belarus in Cork on Tuesday after suffering a calf injury in training.
It is still not clear whether O'Neill will risk including Keane in the 23-man squad for the tournament but the manager insists the LA Galaxy captain is determined to shake off the problem and make the final roster.
"I couldn't say at the moment, but I think he's pretty hopeful," said O'Neill.
"He had it scanned and it's hopefully not too serious, but enough to put him in jeopardy for a while, so we'll see how it is. He had an injection this morning and we'll see how it is when it settles down.
"First of all, I'll get a better update when another 24 hours pass. He had an injection this morning, so he wasn't going to participate in any training today anyway. I'll see how he is.
"Again, Robbie is a pretty quick healer - he says that himself. He is the captain of the team, we would want him to be there and there's not much more I can tell you."