Spurs have won four and drawn one of their opening five games of the Premier League season ahead of facing Arsenal - who boast the same record - to sit second in the table ahead of the north London derby on Sunday.
Kane, who has made a fine start to life since his £120million move to Bayern Munich, says he will never stop cheering on Spurs.
Asked for his thoughts on Spurs' form, Kane told reporters: "Fantastic start - I am always keeping an eye on them and think I will keep an eye on Tottenham for the rest of my life.
"I am really happy to see the team playing the way they are and of course to see their fans happy as well is a great thing.
"I am sure they will be excited to keep that up. They have a massive game coming up, so for sure I will see how they get on over the course of the season."
Kane insisted that Bayern was the right choice of club to leave Spurs for as he prepares to face Manchester United, another club who have long been interested in him.
The England captain is understood to have had no desire to move to Old Trafford this summer.
"Obviously over the summer I know there were some talks between a few clubs in the background, but Bayern were a team I was really interested and excited by," he said.
"There were not too many other discussions once they came in. It was between them and Tottenham to talk and then the deal got done.
"My focus is on here. Manchester United are a great club, a really big club as well. I just decided to come here and am really happy I have."