Tottenham Hotspur finished in third place last season, while Everton ended up in 11th place.
Even though the bookies have Spurs down as fifth favourites to win the league, and Everton even further back, expectations are high around both clubs.
And Bilic does not believe that supporters of either side will be entirely happy with only qualifying for the Europa League.
“Spurs will probably be unhappy to finish sixth,” the West Ham boss said. “Everton, and all that. Therefore when we talk about Champions League, whatever, Europa League, mid-table, it is all very dangerous. Make no mistake.
“I am a confident, over-optimistic person but apart from the top seven clubs on paper everybody is looking forward to 40 points. And that includes Leicester also because we all know what can happen in the Premier League, with Newcastle.”
Tottenham finished last season in third place, so a drop down to sixth would represent a regression.
However, the likes of Chelsea, Manchester United, Liverpool and Manchester City will all be hoping to catch Spurs this time around.
Mauricio Pochettino’s side have added two new players to their squad this summer, as they look to protect their place in the top four.
Everton, meanwhile, have raised expectations after the arrival of Farhad Moshiri as the new major shareholder and Ronald Koeman as manager.
Moshiri is expected to back Everton heavily with funds, while Koeman is hoped to be an upgrade on previous boss Roberto Martinez.