Kane scored the only goal of the game with a curled strike into the bottom corner just before half-time in Spurs' 1-0 Premier League win at Fulham on Monday evening.
It was the England captain's 266th goal for Tottenham, putting him level with Jimmy Greaves as the north London club's all-time highest scorer.
Conte is out of contract at the end of the season, having signed an 18-month deal in November 2021, and Kane - who himself is in the last 18 months of his deal - says he has enjoyed working with the Italian.
"I'm not going talk about the manager's future," Kane said after the victory at Craven Cottage.
"That's down to him. I'm sure he'll have conversations with the club and assess where we are as a team.
"He's had a lot going on, on and off the pitch, so he probably needs time to assess it all.
"From my point of view, I really enjoy working with him and it would be fantastic if he did stay."
Kane, who turns 30 in July, expects further discussions on his own situation in the near future.
"There's not been much talk, if I'm totally honest," he said.
"I'm sure there'll be conversations over the coming months but to be honest I'm just focused on this season and doing my best for the team.
"There's still a lot to play for, trying to get in the Champions League, we have the FA Cup and the Champions League still to try to win. My focus is on that.
"I know there's going to be rumours, a lot of talk and speculation about my future but I'm just focused on what I can do."
Fifth-placed Spurs have struggled since the resumption of domestic action following the World Cup break, winning just two of their six Premier League matches, but Monday night's result lifted Conte's side just three points behind fourth-spotted
Manchester United, albeit having played a game more.
For Fulham, it was a second straight defeat in the league, and Bobby Decordova-Reid was left pondering what might have been had Marco Silva's men made more of their early possession and chances.
"I think we started the game really well and for the first 40 minutes we controlled the game and had chances, and probably needed to put one of them away," the forward said.
"I think a lapse of concentration in the last five minutes has given them the opportunity to score and you can't really give Harry Kane an opportunity because he'll put them away.
"So disappointing, but at the same time we were still in the game.
"But then second half, I think that was their game-plan, to shut out and maybe catch us on the counter-attack and we maybe found it hard to break them down and at the end of it we probably ran out of steam."