Juan Mata or David Luíz are believed to be the two players José Mourinho has offered United in exchange, but United continue to insist the player is not for sale.
The player expressed his "anger and confusion" with United on Tuesday night, as he appears destined to leave the club this summer.
New United manager David Myes said last week: "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 I want as many options as possible."
Even though Moyes has claimed Rooney is "not for sale", his comments at the weekend - as well as those from chief executive Edward Woodward - have further undermined the striker.
Rooney's position has now become untenable at United, and Chelsea's swift offer signals their desire to capitalise on his uncertainty at Old Trafford.
The fact Mata and Luiz have been deemed expendable by Mourinho in such a deal seems particularly significant.
Spaniard Mata and Brazilian Luíz were the two outstanding players of Chelsea's underwhelming 2012-13 campaign.
Mata joined from Valencia for £23.5m two years ago after Luíz signed for the Blues in January 2011 for £21.3m and the duo played pivotal roles in Chelsea's Champions League and Europa League triumphs.
Twenty-five-year-old Mata has been named Chelsea player of the year during his two terms at the club and was included in the Premier League Team of the Year for last season.
Mourinho has spoken of his admiration for Rooney on numerous occasions, most recently last week.
When Rooney requested a transfer from United in October 2010, Mourinho, then Real Madrid coach, cheekily suggested Sir Alex Ferguson "give me a call".
I'd like to buy your old Nokia 3210 phone off of you. I'm willing to offer you my iPhone 5 plus one hundred pounds. Do we have a deal?— Jay (@Jay_in_Essex) July 17, 2013
Anger and confusion are Rooney family heirlooms dating back to 2010: pic.twitter.com/tq9guNLKpb— Football Clichés (@FootballCliches) July 17, 2013
Can't work out what's madder: Chelsea offering to swap Mata/Luiz and £10m for Rooney, or United rejecting it.— Rory Smith (@RorySmithTimes) July 17, 2013