AC Milan’s CEO Adriano Galliani has admitted that the club has received “several” offers for Brazilian forward Robinho.
Jose Mourinho insisted the future of Wayne Rooney is a decision for Manchester United as the tussle for the striker's services began in earnest.
Mourinho was low-key when commenting on the offer in Bangkok on Wednesday, despite United provoking the Blues' ire.
Chelsea confirmed they want to buy Rooney, who is in dispute with United over comments made by manager David Moyes when he spoke to journalists in Bangkok last week.
However, they are furious at suggestions David Luiz or Juan Mata had been offered as makeweights in the deal.
Speaking in Bangkok following the first match of his second spell in charge - a 1-0 win over Singha All-Stars - Mourinho declined the opportunity to fuel the argument.
"I think that this situation is clear - nothing is a secret anymore," he said.
"The official bid is just about a certain amount of money and doesn't involve players.
"We want the player. We made the bid and now we have nothing more to say.
"Now it's up to Manchester United."
Asked if Chelsea had made any other bids, Mourinho said: "No, and we won't."
Asked whether it is now "Rooney or bust", Mourinho said: "yes."
The Londoners are certain United were responsible for the suggestion of a part-exchange, which they insist was not true.
"Chelsea Football Club can today confirm that yesterday it made a written offer to Manchester United for the transfer of Wayne Rooney," said Chelsea in a statement.
"Although the terms of that offer are confidential, for the avoidance of doubt and contrary to what is apparently being briefed to the press in Sydney, the proposed purchase does not include the transfer or loan of any players from Chelsea to Manchester United."
Asked specifically about any proposal to swap a player or players for Rooney, Mourinho said: "For me, no more comments. I have to respect Man United and my colleague David (Moyes, the United manager), to respect even my players from Chelsea.
"The bid is there. The club knows we want the player. The player has to know we made a bid for him.
"Now it's up to them and we respect their decision. I work with my players now, as I've done from day one.
"I trust my players, I'm happy with them and that's what I have to do now.
"We did what we had to do in a proper way, in an ethical way, and we're very happy with our behaviour. It was clean.
"Now I keep doing what I'm doing, working with my players, trying to improve my players, and we'll see what happens. Now he's with Man United, not with us."
There has actually been no suggestion United either confirmed the bid, or put the names of Mata and Luiz into the public domain.
For their part, the Blues believe Mata and Luiz - both still on their summer break after featuring in the Confederations Cup - are key to their long-term plans and also want Rooney to be part of Mourinho's second coming.
It does bring an additional twist to a story that is likely to rumble on for most of the summer.
Last week, Mourinho expressed his admiration for Rooney.
But the offer has been made with the England striker back in Manchester for treatment on a hamstring injury and at a time when he is annoyed at the depiction by Moyes of him being a back-up for Robin van Persie.
So deep is his fury, sources claimed he was "angry and confused" about Moyes' statement, it seems impossible his nine-year stay at Old Trafford can be salvaged.
That United should dismiss the offer so quickly is no surprise.
In recent weeks they have repeatedly insisted Rooney is not for sale, and the idea of selling to a domestic rival is particularly unpalatable.
Yet it is difficult to envisage how United can possibly hold that line right until the transfer window closes.
It increasingly appears Rooney's position is impossible, with United fans likely to turn against their one-time hero.
And for Moyes, it represents a tricky situation given he is attempting to bolster his squad rather than weaken it.
Yet Rooney's reaction to Moyes' comments has been the trigger for a further deterioration in the relationship between club and player.
Though it is thought Moyes did not mean the comments in the way they have come across, Rooney has reacted negatively.
Moyes had said: "Overall, my thought on Wayne is that if for any reason we had an injury to Robin van Persie we are going to need him and I want as many options as possible."
As far as Rooney is concerned, he is at the peak of his career and has no intention of playing second fiddle to anyone, or being reduced to the rank of a squad player.
With four Premier League titles and a Champions League winner's medal to his name, the 27-year-old does not believe he has anything to prove at United.
Yet now he feels as though he has been put on trial by Moyes.
After months of speculation about his future, and no statement of intent to stay from the player, it is now clear that unless there is some major repair work undertaken, Rooney will be leaving.
The manner of United's response is the first major test for Moyes and chief executive Ed Woodward since they replaced Sir Alex Ferguson and David Gill respectively at the beginning of the month.
Woodward left United's tour earlier than previously anticipated to attend urgent transfer business in Europe.