The ultimatum was revealed after Chelsea held the defending Premier League champions in a 0-0 stalemate on Monday night.
Mourinho has already failed with two bids for the England star, who was cheered by United fans throughout a sparkling performance at Old Trafford, and has called on Rooney to state where he sees his future.
When asked if a third bid for Rooney was imminent, Mourinho said: "We have to speak but this club must be a very special club, because in every club in the world when a player wants to leave, they don't support him. They give him a hard time. But they support him all the way.
"We are not so silly as to try to get a player from a big club, a club that doesn't sell what they don't want to sell; we are not so silly as to try to do something if somebody didn't start it.
"The person that started the story has to finish the story. For the good of everyone it is time to finish the story. One way or another he has to say 'I want to leave' or 'I want to stay'."
It has been reported that Chelsea have players such as Anzhi Makhachkala's Cameroon striker Samuel Eto'o on their radar and when asked about how long Mourinho was willing to wait before moving onto other targets, the Portuguese said: "Twenty-four hours, 48 hours."
United manager David Moyes has insisted throughout the summer that Rooney would not be sold and after Monday night's performance his stance is unlikely to change.
In a match of few clear-cut chances, the 27-year-old England forward was instrumental in most of United's moves and looked fully determined to focus on activities on the pitch.
Moyes said on Monday night it would be "illegal" for Mourinho to make any request of Rooney and hailed his striker's performance, as well as the fans.
The Scot said: "I was very pleased with his performance and I was very pleased with the reaction from the crowd.
"I think Wayne has a positive mind at the moment. I don't think he needs to change. I think his performance said enough. I think his actions showed what he thought."
Mourinho added: "Rooney was a real professional. He played for his club, played for the club that pays him, tried to win, tried to score. He was fantastic.
"After that, if he wants to leave, he has to say - or he decides now that he doesn't want to leave anymore. But I praise him. He played a fantastic match in these difficult circumstances.
"They gave him fantastic support, it was good, it was nice - very English."