Kane's first-half strike in Sunday's 1-0 win against defending champions Manchester City took him past Jimmy Greaves to a record 267 goals for the north London club.
The 29-year-old's predatory finish also made him only the third player to reach 200 goals in the Premier League, behind record holder Alan Shearer (260) and Wayne Rooney (208).
Now his place in Tottenham history is secured, Kane admitted he would love to break Shearer's record as well.
"I'm sure you guys will be talking about that," he said. "It's always there in the back of my mind but I just take it game-by-game, season-by-season.
"It's definitely there to be broken and I'm feeling good and feeling fit so we'll see what happens."
Kane's remarkable goal feats are all the more incredible given he was regarded as unlikely to make it at the highest level at the start of his career.
After graduating from Tottenham's youth academy, Kane was sent on a series of loan spells in the lower leagues before finally being given his chance to shine in the Premier League.
"It's been an amazing journey," he said. "I've been here since I was 11 years old, it's 18 years of my life, a lot of hard work and dedication.
"I'm extremely proud to represent this club and score 267 goals for them and 200 Premier League goals for them is something I'll never take for granted.
"I've just got to keep it going, score as many as I can and see what happens from there."
Kane could make further history later this season as he looks to become England's record scorer.
The England captain moved into joint first place with Rooney when he netted his 53rd international goal in the World Cup quarter-final defeat against France in December.
He could have broken the record in the same game but blazed a late penalty over the bar.
Kane's next chance to pass Rooney is likely to come in England's European Championship qualifiers, with games against Italy and Ukraine scheduled for March.
For now, Kane is basking in the afterglow of his Spurs landmark, with boss Antonio Conte congratulating him by phone after the City game.
Conte underwent surgery to remove his gallbladder last week and watched the win on TV while he recovers in Italy.
"He was on the phone there and he just kind of congratulated me for the achievement and just said he was proud of all the boys," Kane said.
"It's not easy when you don't have your coach there. It's obviously been a difficult week. We're all happy the surgery went well and he's recovering well.
"We showed we're a unit, a team. I'm sure he was watching, proud. It was the type of performance where we knew what we had to do."
Tottenham are just one point behind fourth-placed Newcastle after their win, with second-placed City five points adrift of Premier League leaders Arsenal.
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