Kane bemoans another trophyless season for Spurs
Spurs drew 0-0 with AC Milan at home on Wednesday to lose their last-16 tie 1-0 on aggregate.
That followed a dismal FA Cup defeat to Championship side Sheffield United and a loss at Wolves that weakened Tottenham's grip on fourth place in the Premier League.
Defeat against Milan ensured Tottenham will complete a 15th season without silverware.
Kane, who broke the club's all-time goalscoring record earlier this season, was adamant that Spurs need to be doing more than just challenging for the top four, which ensures qualification for the Champions League.
The 29-year-old England captain is out of contract at the end of next season and realistically has one last chance to sign for an elite club that will guarantee him the chance to win regular trophies.
Asked whether a top-four finish is enough, Kane said: "I don't think so. Where we're at as a club, we should be winning trophies. That's always the aim.
"The top four (being our only target) is a consequence of not playing as well as we want to play. Now that's all we can fight for so that's going to be the goal and hopefully we can achieve that come the end of the season.
"But for sure, it's not enough for this club."
Kane said the FA Cup defeat to Sheffield United had "put a dagger in our hearts", halting their momentum.
Spurs' meek exit from Europe could speed up Antonio Conte's expected departure from the club -- the Italian is out of contract at the end of this season.
But Kane said all the players could do was give 100 percent for their manager.
"That's his decision, he's going to be the one that makes that choice," said the forward. "All we can do as players is try to perform for him, work as hard as we can.
"That's what we're doing. I said after the Wolves game at the weekend, you can't fault the effort of the players.
"We're trying, we're training but we're just lacking something. We've talked about mentality before and that ruthless hunger to be better, to be the best, to be one of the best teams in Europe.
"We just haven't quite found that yet."