Kane has established himself as one of the best No.9s in the world by scoring 140 goals in all competitions for Tottenham since making his debut for the club in 2011, while he will also captain England at the World Cup in Russia.
He has repeatedly been linked with a move to Madrid, with former Spurs star Paul Gascoigne claiming the club would not reject a £100 million (€113m/$133m) offer for the 24-year-old.
Though Spurs will be determined to keep hold of Kane through the summer, Pavlyuchenko feels it would be hard to pass up the opportunity to join one of the world’s biggest clubs.
“But, he must first know what he wants. It is certain that, if Barca and Real are present, then it's normal to want to go there.
“Tottenham is a big club, but they don’t play in the final stages of the Champions League. If he wants to go even further, he must, of course, join a bigger club.”
Kane played alongside Pavlyuchenko in the early stages of his Spurs career before the Russian left White Hart Lane in the 2012 January window to join Lokomotiv Moscow.
Pavlyuchenko has been pleased to see the progress Kane has made in recent years, saying: “Today, as you can see, he scores a lot of goals. Many clubs want him and for that I can only say “well done”.
“I look at him and see how effective he is. This pleases me, especially since I was at his side during the first steps. I am very happy for him. He is a great Tottenham player today.”
Next season, Spurs will be playing in the Champions League for the third year in a row and Pavlyuchenko believes the club’s enduring style has helped with their recent success.
He added: “Tottenham has always had a very seductive style of football. This team is very attacking and produces a good passing game.
“They play a spectacular type of football. The players come and go, but the style remains the same.
“Personally, I have always liked it. When Tottenham became interested in me, I always said that this style suited me. And, it seems to me today that this style has endured and has not changed much.
“When Tottenham became interested in me, I always said that the style suited me, and it seemed to me today that this style has endured and has not changed much."