The 32-year-old England captain missed his country's 1-0 friendly defeat by Japan last month due to the problem and was also absent for Bayern's 3-2 win at Freiburg.
Kane's potential return would be a major boost for Bayern, with the prolific forward having scored 48 goals in 40 appearances across all competitions this season.
The six-time European champions will be eager to have him fit as they prepare to face a Real side who cruised into the quarter-finals with a dominant 5-1 aggregate victory over Manchester City.
"It was important that he did a lot of training today. I don't think he's lost his rhythm. We'll wait until tomorrow and then decide," Kompany told reporters on Monday.
"Everyone is available apart from (goalkeeper) Sven Ulreich. We have a strong squad and we're ready for the two legs."
Bayern arrive in Madrid in red-hot form, sitting nine points clear at the top of the Bundesliga and having finished second in the Champions League phase, suffering one defeat.
Real, meanwhile, are looking to respond after a 2-1 LaLiga loss at relegation-threatened Mallorca on Saturday.
"The most important thing is that we have full focus on the game tomorrow. We want to show what we're capable of. We'll give our all to do that. I just want us to win and not have any fear," Kompany said.
The former Belgium and City centre back acknowledged the scale of the challenge facing his side but said Bayern are ready.
"Everything will matter tomorrow. This is my first time here as a coach, but we have a lot of experience in games like this," he said.
"The outcome will be decided in part by tactics, but above all by the little things - such as the individual quality of the players. We've prepared for Real and we've seen what they did against City. It's perhaps the toughest away game you can play, but we want to win."