Long-serving keeper Cech is set to battle Thibaut Courtois for the No.1 spot in the new season with Mourinho bringing the Belgian back from a successful loan stint at Atletico Madrid.
Cech, who missed the back end of last season with a dislocated shoulder, made his return to action in the 1-1 pre-season draw with Wolfsberger in Austria on Wednesday and he is determined to keep his spot in the team ahead of Courtois.
"I don't want to be on the bench," said Cech. "This is why I am trying to do everything to be in the team.
"I don't want to talk about ifs and buts. Let’s see how I am going and what happens.
"You know I always give my best and obviously that is all I can do. I will do my best and do everything I can do to be as ready as possible for when the season starts.
"Every year there is a challenge. If you are going to be in the team you are going to have to do better than the other guys in the team. I think Mark Schwarzer, when he came last year, did remarkably well during the pre-season and when he played in any game he did very well, so I knew the challenge was there."
Cech insists he is doing everything he can in pre-season to impress Mourinho as he looks to fight off the formidable challenge of Courtois.
He added: "Obviously Jose Mourinho will make his choice and decide how his team is going to look for the first game, second game, third game, whatever game.
"It is up to us players to show we are ready to play and make him pick us. I am not any different. I am doing everything I can to make him pick me."