The Blues have already seen two offers for the England international knocked back by the Red Devils.
United manager David Moyes maintains that Rooney is not for sale, and drafted him back into his plans with a substitute run-out at Swansea City on the opening day of the new Premier League season.
There are, however, still two weeks until the summer transfer window closes and United's resolve could be tested by a third bid from Stamford Bridge.
Mourinho has stopped short of confirming that Rooney will be the man he goes for before deadline day, but admits bolstering his attacking options remains a top priority.
"We will try till the last day to add a new player to the squad, a striker, but in this moment every striker has a club, every striker belongs to somebody and I don't think it's ethical that I name players that belong to other clubs," Mourinho said following Chelsea's 2-0 victory over Hull City on Sunday.
"We don't try, like many clubs do, to go around and try to influence players to behave in a certain way.
"If we have to make a bid, we make it in an official way, we don't speak to players, we speak to clubs."
While Chelsea continue to monitor Rooney's situation at Old Trafford closely, those inside the United camp are hoping the 27-year-old can be persuaded to extend a nine-year association with the club.
Midfielder Michael Carrick said: "Wayne's a big United player. He's one of us and we all stick together. When we're on the pitch we fight for each. He's part of that.
"I thought he came on and looked good (against Swansea). He's a real threat.
"He's given this club an awful lot, so the fans support him and we support him."