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The England captain says Spurs' fate is still in their own hands given they host their rivals in the north London derby next month, but thinks they will have to win all five of their remaining games this season to seal the final Champions League place.
For the second consecutive game - and a league-high fourth time this season - Tottenham failed to register a shot on target during their goalless draw at Brentford, having also drawn a blank in the defeat by Brighton last weekend.
Spurs stepped out in west London on Saturday evening with Arsenal having put the pressure on with their 3-1 win over Manchester United in the early top-flight kick-off.
Antonio Conte's side could not respond and ended the day two points adrift of their fierce rivals, who visit the Tottenham Hotspur Stadium for a mouthwatering derby on May 12.
"To only get one point from those two games [Brighton and Brentford] is disappointing," Kane told Standard Sport. "We're running out of games in terms of dropping points.
"There are five games left and we feel like we are pretty much going to have to win all of them to get that spot. We are more than capable of doing that, we believe in that and that is what we'll try and do."
Kane may be confident that this inconsistent Spurs side can end the season with five straight wins, but it is a tall order. After hosting Leicester this weekend, Conte takes his team to Anfield before the north London derby. Tottenham end the season against Burnley and Norwich.
"Is it still in our hands? Yeah," said Kane. "If we win the last five games we will be in the Champions League, so that is how we've got to look at it.
"We know those five games will be difficult, we go to Liverpool away as well which will be very tough.
"As we've seen this season, we have dropped points against teams where we were probably expected to win and so have the others around us. It is by all means not done yet.
"I think we've done great to put ourselves in this position we're in now but it is about who can finish it off.
"We have to stay focused and keep working hard. We have a little gap now to prepare and work and hopefully we can put in a good performance against Leicester."
Conte waited to make changes against Brentford despite Tottenham's lack of cutting edge. When a substitute was finally introduced, the Italian sent on Davinson Sanchez to shore up his defence rather than liven up the forward line. Conte only turned to Lucas Moura with four minutes to go in the match.
"Don't forget sometimes the most important thing is to continue to have a good balance," said Conte, whose side almost lost it at the death when Ivan Toney headed onto the post. "Because otherwise if you want to go only to win then sometimes you risk to lose the game."