Kane picked up a hamstring strain during Tottenham's convincing win over Liverpool last weekend, and was absent for their defeat at Manchester United on Saturday.
Spurs boss Mauricio Pochettino insisted the 1-0 loss at Old Trafford was not down to Kane's absence, and the in-form striker needs to be assessed before their fourth Champions League Group H tie.
"We will see in the next few days if he will be available and fit to play against Real Madrid," Pochettino said. "We need to assess him tomorrow, and after tomorrow, and then make a decision.
"I told everyone before the game that, of course, we would miss our main striker - he is one of the best strikers in the world.
"But that is not an excuse today. We didn't lose because Harry wasn't here. It is a collective sport and you must always adapt when some players are not fit to play. "
Tottenham will hope they won't be without their talisman for the crucial clash against Madrid, as he has been in unstoppable form this season, with 13 goals in 12 games for his club.