Former Everton manager David Moyes lodged a 28 millionpounds (USD 44 million, 33 million euros) bid to bring the pair to Old Trafford, including 16 million pounds for Fellaini, the BBC and other media reported. A 23.5 million pounds release clause in Fellaini's contract expired at the end of July.
The Premier League champions are believed to have offered the same amount for 28-year-old Baines -- 12 million pounds -- that they bid in June when the offer was flatly rejected.
The BBC website said Everton had greeted the new bid for Baines with anger. Everton claim Moyes made an unwritten pact with the club not to try to sign their players when he left in the summer to succeed Alex Ferguson.
Moyes signed both players for Everton, paying Wigan 6 million pounds for Baines in August 2007 and signing the lanky Fellaini for a then club record 15m pounds from Belgian club Standard Liege in September 2008.
Moyes has been frustrated in the transfer market with attempts to bring Barcelona midfielder Cesc Fabregas to United so far proving unsuccessful, while England striker Wayne Rooney appears to want to leave Old Trafford for Chelsea.
United crushed Swansea 4-1 in their opening match of the Premier League season on Saturday, while Baines and Fellaini both played as Everton drew 2-2 with Norwich.