"The person that starts the story has to finish the story," Mourinho said Monday after Chelsea's 0-0 draw in the Premier League against a United side that contained Rooney.
Rooney indicated to United toward the end of last season that he wanted to leave, and the English champions have turned down two bids from Chelsea for the striker. United's stance all summer has been that Rooney is not for sale.
"If now he does not want to leave, we are out of the scene," Mourinho said, "but we need to know what is happening."
"We are going to (sign) another player if he is not coming," added Mourinho, who has also been heavily linked with a move for Anzhi Makhachkala striker Samuel Eto'o.
When asked directly if he wanted Rooney to come out and make a declaration about his future, Mourinho said: "Yes," adding that he needed an answer in "24 hours, 48 hours."
Rooney played the full match against Chelsea in his first start of the season and was given superb encouragement by United's fans, despite having officially asked for a transfer request in 2010 before again questioning his future at the club in the final days of Alex Ferguson's reign last season.
"This club must be a very special club because at every club in the world, when the player wants to leave they don't support him," Mourinho said. "When a player wants to leave they give him a hard time but (here) they support him all the way, so I think this is a real special club with special fans so he decides that he wants to stay. If that is his decision then we will be the first to respect that."
Mourinho said Rooney played like a "real professional" and was "fantastic."