Kane, who did not have a single shot in the game against Leverkusen, was injured in a collision late in the match at the Allianz Arena and was in obvious pain before he limped off the pitch.
Bayern are top of the table in the new Champions League format having beaten Dinamo Zagreb 9-2 in their first game where Kane scored four times. Villa are fourth in the standings.
"I'm not a doctor, but I hope it's nothing serious. It's still early to say. Hopefully he'll be fit for Wednesday," Kompany told Sky Sports after the game.
Bayern sporting director Max Eberl was also not optimistic, saying the injury "doesn't look good" for Kane, who leads the Bundesliga scoring charts and has 10 goals in all competitions this season.
"If Harry goes off the pitch, then it's something," Eberl said. "Now I hope he's made of sturdy English wood."
Bayern CEO Jan-Christian Dreesen was a little more hopeful, however, saying: "He took a knock, but as the doctors have said, it's probably nothing serious. We'll have to look at it again tomorrow."
Leverkusen made things difficult for Bayern with a superb defensive performance to end Bayern's perfect start to the season but Kompany still saw positives from the game.
"As a Bayern coach, you always want to win. But the performance was as we'd hoped. The counter-pressing was positive, we won the second balls, we created so many chances," the Belgian coach said.
"Normally in such a game, you don't create many chances, but we did. It was a top game but there are still many games to go. We just have to keep going."