Cech's days as Chelsea's first-choice stopper appear to be numbered.
The 32-year-old, who has been Chelsea number one for 10 years, had to settle for a place on the bench on Monday night as Mourinho opted to field Belgian goalkeeper Thibaut Courtois in the first match of the season against Burnley.
Mourinho confirmed Courtois would keep his place in the team after an impressive performance in the 3-1 win, which came courtesy of goals from Andre Schurrle, Diego Costa and Branislav Ivanovic.
But the Chelsea boss insisted Cech is capable of winning the jersey back if he stays at the club.
"Yes, I want to keep (him)," Mourinho said.
"The club wants to keep him and hopefully he stays.
"Courtois was my first choice today and he played good enough to be my first choice in the next game.
"He is not my first choice for the rest of the season.
"When you have a top professional working hard like Petr, if you don't perform you are in trouble."
Paris St Germain and Monaco are said to be two of the parties interested in Cech's signature.
Mourinho admitted he has a hard task on his hands keeping Cech happy, but he thinks the goalkeeper will not give up without a fight.
"No player is happy when he is not playing every game," he said.
"But I know Petr for 10 years, I know he will not relax, he will not sit in a chair and say 'I have no chance'
He will be exactly the opposite
I am expecting him Wednesday to go to the training ground and fight."
When it was put to him that Cech could request to leave, Mourinho said: "He didn't ask and I hope he doesn't ask."
Source : PA
Source: PA