The Hammers set up home at the London Stadium last summer, but they failed to build on last season's seventh-placed finish, winning just seven of their home games.
Tottenham will move into their new stadium for the 2018-19 campaign, but next season they will play their home matches at Wembley, where they struggled in the Champions League and Europa League.
Despite winning just one of their five games at the stadium, Kane is not concerned that they will suffer like West Ham.
"I think the way we play is a different style to West Ham," The Independent quotes Kane as saying. "When we're out there, we're going to try and play exciting football, press high like we do here and that should get the crowd on their feet.
"The crowd have always been good, even in the Champions League when we played at Wembley the crowd were fantastic. It was just our performances on them days weren't great. I am not too worried, I don't think the players are too worried.
"We're just excited to take the next step, take the next challenge and that is going to be even more difficult next year."
Spurs have sealed their highest-ever finish in the Premier League by clinching second.