With Chelsea not in action until Monday's clash with Middlesbrough, Mauricio Pochettino's second placed side can temporarily move within one point of top spot if they beat the Hammers at the London Stadium.
Tottenham know the affect the schedule can have after they regularly followed Leicester during last year's run-in and finally crumbled to hand the Foxes the title.
So far, they have coped admirably with the anxiety of having to play their fixtures after Chelsea, responding to victories by the leaders by beating Crystal Palace and Arsenal in their last two games.
This weekend, Kane and company, winners of their last nine league games, go first and the Tottenham striker is determined to seize the chance to turn up the heat on the Blues.
"We just have to try to finish strong. We've got another game Friday night and it's good to play first," Kane said.
"Hopefully we can drop the gap to one point and then see what happens.
"We know what it's like from when Leicester were playing last year before us, and even today, that it's good to play first and put the pressure on.
"It'll be a tough game away from home but hopefully we can get that win, put the pressure on and just wait and see."
Chelsea maintained their charge with a 3-0 win at Everton on Sunday before Tottenham brushed aside Arsenal with a 2-0 victory.
Kane's penalty helped seal success in the last north London derby to be played at White Hart Lane.
Tottenham will play at Wembley next season while their new stadium is completed and Kane is certain his emerging young team are destined for great things no matter where they call home.
"Every time we drive in as a squad now you can see how big that stand is getting," Kane said.
"Everything is getting built up and it shows the way the club is going.
"We've got a fantastic manager, he's one of the best in the world in my opinion, we've got a great squad with a real togetherness.
"We're just going in the right direction. We can't get carried away, even this season we've got four games left to try to win and it's exciting times."
Tottenham's win last weekend means they are guaranteed to finish above Arsenal for the first time since 1995, but Kane is far more concerned with landing the title.
"It was an added bonus that in the last game at the Lane we managed to finish above Arsenal," Kane said.
"It's great for the fans and great for the club. But we're thinking about trying to get that title and we'll see what happens."
West Ham would relish spoiling Tottenham's title ambitions as they seek to secure their place in the top-flight.
There is little love lost between these two London rivals and police will be on high alert to ensure there is no repeat of the violence that marred several of West Ham's first fixtures at their new home.
With his side seven points above the relegation zone, Hammers boss Slaven Bilic had demanded a strong finish to move into mid-table in their remaining games against Tottenham, Liverpool and Burnley.
"We need more points, it may be enough," Bilic said.
"We don't want to come into the last game of the season with the radio on and keep asking 'what is the score there?'
"We would love to avoid it. We are not panicking but we are not safe.
"There are nine points to play for and we need to pick up more."