The Blues are involved in a fierce three-way battle for Champions League football alongside London rivals Arsenal and Tottenham Hotspur, with the trio separated by just two points.
Cahill has declared that the Blues must qualify for Europe's elite club competition and maintain high standards in the search for silverware.
He told The Telegraph: "There's a little bit of fear there now. A club like Chelsea expects to be in that competition.
"Of course that's what we have to do. It's massively important for the club and players and we are all well aware of that.
"You come to a club like this and the demands are so high, they expect to do things, to win things and that rubs off on you. Over the next few years I want to be a part of winning a lot of trophies. It gives the players a hunger."
The 27-year-old is battling to be fit for Thursday's Europa League semi-final first leg against FC Basel as he continues to recover from minor knee surgery.