Kane bagged a brace as England beat Tunisia 2-1, scored a hat-trick as his side thrashed Panama 6-1, and netted one goal for his side as they defeated Colombia on penalties.
Three of the six goals were scored from the penalty spot.
"He [Kane] captains the team exceptionally well, he led the culture we're trying to create," England head coach Gareth Southgate told journalists.
France forward Antoine Griezmann is the runner-up, having scored four goals, Belgium forward Romelu Lukaku is in third place as he scored four goals as well, but Griezmann made more assists.
Midfielder Denis Cheryshev is the top goalscorer of the Russian national team, as he netted four goals for his side during their home tournament.
England lost to Belgium 2-0 in St. Petersburg earlier on Saturday, matching their result of finishing fourth in 1990.