The England forward is apparently "angry and confused" about his status at United, following recent comments made by new manager David Moyes and chief executive Ed Woodward.
And, after Chelsea boss Jose Mourinho spoke last week about how much he admired Rooney as a player, the London club have now placed a firm bid for the 27-year-old.
It is believed the Blues have offered £10m plus either Spanish playmaker Juan Mata or Brazilian defender David Luiz.
However, since taking over at Old Trafford, Moyes has stressed that Rooney is not for sale, and the bid has been swiftly declined.
Rooney is currently being treated for a hamstring injury at United's training ground, having had to fly home from the club's pre-season tour.
Sky Sports News reporter James Cooper said: "Various sources have confirmed that Chelsea have made an opening bid for Wayne Rooney.
"Manchester United told them to go away rather abruptly and told them also that Wayne Rooney is not for sale. That's the mantra the club have been sticking to since this situation began.
"If they want to get rid of the player, they want to make sure that his value is protected and, by saying he is not for sale, it means the games can begin."
Rooney has enjoyed a hugely successful nine-season stint at United but was alarmed by recent public remarks about his role at United from Moyes, who said: "If, for any reason, we had an injury to Robin van Persie, we'll need him.
"I've got to ensure it's not just him and we don't concentrate on him. Manchester United isn't about Wayne, Manchester United is about the team, the club."
Rooney is also unhappy that Woodward has categorically ruled out re-negotiating his contract, which still has two seasons to run.
Speaking last week, United chief executive Woodward said of Rooney's position: "No contract renewals are being discussed. I am not sitting down with any player on an extension and there is no trigger date in the diary.
"Would we be afraid to run a contract down? Of course not."
Former United centre-half Gordon McQueen believes Rooney will now leave Old Trafford, but thinks it would be a big mistake to allow him to join a Premier League rival like Chelsea or Manchester City.
"It would be a big move for Wayne Rooney and one that I hope doesn't happen," McQueen told Sky Sports News. "I hope he stays with Manchester United. Unfortunately it seems to me that the player has got his mind made up that he wants to leave Old Trafford and it looks like to me now that they will probably let him go.
"I think they would prefer him to go to one of the teams in Europe that can afford him - Paris St Germain or Monaco - you certainly wouldn't want him to go to Chelsea or Man City!
"Man Utd did something similar last season and, in my opinion, it won them the league. That was getting a terrific goalscorer from one of their main rivals, Robin van Persie from Arsenal.
"I think it would enhance Chelsea's chances of winning the Premier League if they took Wayne Rooney from Manchester United - just look at the goals he has scored for Man United over the last nine years. They (United) would find that difficult to replace."
Chelsea emerged as favourites to land Rooney after Mourinho confirmed last week that he was a fan of the striker.
Speaking at the start of Chelsea's Far East tour last Friday, Mourinho said of Rooney: "He is a player I like very much but I can say no more than that. Being fast and direct, I like that, but he's a Manchester United player."
Mourinho insisted earlier this week that he was happy with the squad already at his disposal at Stamford Bridge, playing down talk of any departures.
He has Fernando Torres, Demba Ba and Romelu Lukaku as attacking options, and has bolstered his squad with the likes of Andre Schurrle and goalkeeper Mark Schwarzer.
Rooney has been backed into odds on to make the move to Stamford Bridge. Sky Bet have cut the odds on the striker moving to Chelsea from 11/2 to 1/2, while the odds on him remaining at Manchester United have drifted out to 13/8
Sky Bet Football Trader Chris Spicer said: "We were convinced that Rooney would not be allowed to leave after Moyes laid down his initial stance, but his recent comments are somewhat confusing.
"Relations are starting to appear irreparable and we have now installed Chelsea as the favourites for the Premier League at 9/4, which would shorten significantly if they secure Rooney's services."