United snubbed the Blues' opening £23m offer last month and despite Rooney's wish to end his nine years at Old Trafford, and the majority of supporters' keenness for him to leave, the Premier League champions have not softened their stance.
David Moyes has clarified on a number of occasions Rooney is "not for sale", but the striker, "angry and confused" about his new manager's comments about his role in the squad, is determined to leave.
Rooney scored against Chelsea in the FA Cup last season
A United spokesman said: "A bid was received yesterday and immediately rejected. Our position remains that he is not for sale."
Rooney played in a behind-closed-doors friendly win against Real Betis on Saturday but has not appeared in any official pre-season matches for United.
However the 27-year-old could gauge supporters' feelings about him this week, when United play their final three pre-season fixtures.
The club travel to Scandinavia to face AIK on Tuesday, entertain Sevilla in Rio Ferdinand's testimonial on Friday and contest the Community Shield against Wigan Athletic at Wembley on Sunday.
It is expected Rooney will submit a written transfer request later this week.
Ferguson frostily congratulates Rooney ahead of receiving the Premier League trophy
The Croxteth forward's relationship with United has been uneasy the past few seasons since he asked for a transfer in October 2010.
Although Sir Alex Ferguson convinced Rooney to sign a new five-year contract instead, the duo's relationship never fully recovered.
Ferguson fined and dropped Rooney in December 2011 for going out drinking and even referred to the striker as "stocky" last year, amid accusations Rooney was unfit at the start of the 2012-13 campaign.
Demoted for the return leg versus Real Madrid in the Champions League, the Scot later revealed after his final Old Trafford match in May Rooney had asked to leave United, something the striker denies.
Rooney celebrates a vital winner at Fulham in February
Marginalised by Robin van Persie in United's 20th title-winning campaign, Rooney nevertheless struck pivotal winners against Southampton, Fulham and Reading, as each victory extended United's points advantage over neighbours City.
The former Everton player was also the prime victim of Ferguson's fondness for tinkering. He has played in at least five different positions for United during his Reds career.
Chelsea have identified Rooney as the ideal striker to replace Didier Drogba after Fernando Torres' hapless attempts over the last two-and-a-half-years.
José Mourinho, who spoke of his admiration for Rooney as Real Madrid boss three years ago, currently has Torres, Romelu Lukaku and Demba Ba to choose from as the Blues' main striker.