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British reports earlier said the Premier League champions had rejected Chelsea's offer for the disaffected England striker.
"Chelsea Football Club can confirm that yesterday it made a written offer to Manchester United for the transfer of Wayne Rooney," said a club statement.
British media said Chelsea had made a £10 million, with either midfielder Juan Mata or defender David Luiz thrown in.
The club statement did not reveal the price offered for Rooney but it said no player had been offered as part of the deal.
"Although the terms of the offer are confidential, for avoidance of doubt and contrary to what is apparently being briefed to the press in Sydney, the proposed purchase price does not include the transfer or loan of any players from Chelsea to Manchester United," the club said.
Chelsea manager Jose Mourinho, who is searching for a new striker, this week said he admires Rooney "very, very much" but stopped short of saying he wanted to sign the England forward.
Manchester United boss David Moyes meanwhile has repeatedly stressed that Rooney, who is said to have asked for a transfer in May, is not for sale.
"It's hard to keep repeating myself but I will do -- Wayne's a Manchester United player and will remain that," Moyes said last week.
Rooney, however, was on Tuesday reported to be "angry and confused" by later comments from Moyes that he would be used as back-up to striker Robin van Persie next season.
