The Spurs forward, who scored on his England debut last month against Lithuania, reached the 30-goal mark for the club with his strike in the 3-1 win at Newcastle United on Sunday.
'I need to set myself another target now - but I will keep that quiet'
Harry Kane
That moved him ahead of Jurgen Klinsmann’s 29 goal haul in the 1994-95 campaign, and the hitman is closing in on Lineker’s total of 35 from 1991/92.
Following his strike, the 21-year-old said: “I’m very proud to reach 30, but there are five games left and hopefully I’ll get a few more goals. I had a couple of chances and the keeper made some good saves but I’m delighted to get back on the scoresheet.”
Speaking to BBC Sport, Kane added: “After I got to 20 goals, I set myself a target of 30 and it felt like I was on 29 for ages.
"But now I've got here I'm delighted. Maybe I need to set myself another target from now to the end of the season [but] I will keep that quiet.”
Despite the prospect of his record being broken, Lineker was quick to offer his congratulations to Kane on Twitter.
Well done @hkane28 on being the first Spurs player to score 30 goals in yonks. Sheringham, Klinsmann, Defoe, Keane, Adebayor, where are you?
— Gary Lineker (@GaryLineker) April 19, 2015
Having seen his young marksman reach the impressive milestone, Spurs boss Mauricio Pochettino joked that the Kane should buy his team-mates dinner.
"Congratulations to him because he deserves this moment,” Pochettino said.
"I think he needs to pay for a dinner this week for his team-mates. The team work a lot and try to help him score."