United have maintained that the 27-year-old is not for sale, but Mourinho said following a pre-season loss to Real Madrid in Miami on Thursday that they would continue their pursuit.
"We are far away so we are not in England to know every detail but we are doing things in the correct way," Mourinho told reporters in America.
"We made the bid and Manchester United have reacted. Let's see what happens. But we can't do more than we are doing now.
"We are doing things legally, making the official bid directly to the club, no interviews or comments or direct relations with the player, no contact, absolutely nothing.
"We are doing everything in an ethical way, so let's see if things change," added Mourinho, who was fined by the Premier League in 2005 for tapping up Ashley Cole prior to the defender's transfer from Arsenal during his first stint in charge of the Blues.
Mourinho also said he had no interest in selling Brazilian defender David Luiz to Barcelona and was happy to be patient over Rooney, with Chelsea already boasting a strikeforce of Fernando Torres, Romelu Lukaku and Demba Ba.
"There is no time limit," Mourinho said. "We have our squad of strikers. We have identified the player as one we would like to have with us. We have done it in an ethical way and we are going to do that until the last day."
ENGLAND FOCUS
Rooney has yet to appear for United in pre-season, having suffered hamstring and shoulder injuries, and his possible departure would leave the Old Trafford side short up front.
United manager David Moyes has yet to make a significant addition since taking over from Alex Ferguson, with hopes of landing Barcelona's Cesc Fabregas squashed on Thursday when the former Arsenal midfielder said he was not planning to leave the Nou Camp.
Despite training with United's reserves to improve his fitness, Rooney was selected by England manager Roy Hodgson on Thursday for next week's friendly against Scotland.
Rooney will join Chelsea players Cole, Gary Cahill and Frank Lampard in the England squad but Hodgson told reporters on Thursday that he would instruct his players not to discuss club matters once they assembled on Sunday.
"I do accept a duty of care," he said.
"We're not naive. Ashley Cole and Wayne are quite close friends. David Moyes must know that as well. If they weren't speaking in a hotel, they would be speaking on the telephone."
"I will try to make it clear to everybody, the Chelsea players and the Manchester United players and anyone else for that matter, that this is England.
"As far as we're concerned, you can speak as much as you like to each other now but during these two or three days this is England, this is England against Scotland and, even more importantly, the matches beyond Scotland."
(Reporting by Josh Reich. Editing by Patrick Johnston)
