Prop Oliver Tomaszczyk (hamstring) and lock James Percival (hand) are also likely to miss the clash with Bath Rugby as the coaching team at Warriors now examine the options in the pack.
"We are probably not going to see Kai much before Christmas because of injury, which is a disappointment to us all," admitted Hill.
"Oliver did a hamstring over in Parma and Adam has also torn one which looks like it could possibly be bad, but he will have a scan later in the week so we must wait and see.
"With the likes of James Percival also out having had a minor operation on this thumb that will keep him out for a few weeks, we are a little bit thin. We are going through a period when, unusually for us, we are getting injuries and not small ones.
"We are one injury away from having to take emergency action to get somebody in," he added.
"We have got Richard Muagututia as a back row option, but he is due to go to Canberra to renew his visa. He came to us on an initially three-month one and it was always the understanding that he would go back and renew it until the end of the season.
"Because of that, we are in the process of looking and I have got in touch with agents to see who is available. It is quite a healthy list so if we had to go with it then we could get somebody in at very short notice."
While Hill admits his forward pack has taken some hits he revealed the strength in depth in the backs was giving him plenty of selection options.
He said: "Jonny Arr is back and sharp and Josh Drauniniu has worked his way to first team level, so has Tom Arscott.
"All the wingers are in good form and Chris Pennell has also worked his way back to being a Premiership player. He has come through his injury spell and I have no doubt that we could start all these players."