The 24-year-old opened the scoring for England against Tunisia in Monday night's Group G match inside the first 11 minutes before Ferjani Sassi netted an equaliser from the penalty spot.
As the minutes ticked by, the Three Lions were heading towards a draw before Kane nabbed his second goal of the night one minute into stoppage time.
The forward has been described by many as world class, but Southgate wants more from his star man.
"I'll keep challenging him because I'm always a little uncertain as to what that title means and what you need to do to really gain it, but he's a player we are privileged to have and I'm delighted to have him as my captain," the England boss told reporters.
England's next match will be against Panama on Sunday.