The England international, who has also attracted interest from Arsenal, is expected to leave Manchester United this summer after handing in a transfer request at the end of last season.
The Blues have already seen two offers rejected for the striker, and Mourinho feels that Rooney has to let his feelings known before Chelsea go in pursuit of other targets.
"The person that started the story has to finish the story. For the good of everyone it is time to finish the story," Mourinho told reporters. "One way or another he has to say 'I want to leave' or 'I want to stay'."
Asked how long he was prepared to wait before moving on to other targets, Mourinho replied: "24 hours, 48 hours.
"If you look at a manager like me, a club like us and the people who work at the club with me, we are not silly enough to try and get a player from a big club that doesn't want to sell. We are not silly enough to try something if somebody didn't start it."
Rooney started United's 0-0 draw with Chelsea at Old Trafford on Monday.