The pair have been consistently linked with a move to United since former Toffees boss David Moyes replaced Sir Alex Ferguson at Old Trafford, but Everton remain determined to keep hold of their prize assets.
The Liverpool Echo reported that United chief executive Ed Woodward tabled the bid, valuing Fellaini at £16m and Baines at £12m just 48 hours before the start of the new season.
An Everton statement, written by Director of Communications Alan Myers, said: "Everton can confirm that it has received bids from Manchester United for two players. Those bids were immediately rejected out of hand as derisory and insulting.
"The club did not make public these details as it was vital Roberto Martinez’s preparations for the opening game of the Premier League season were not disrupted in any way."
United’s valuation of Fellaini infuriated the Goodison club, considering that a recently-expired buy-out clause in Fellaini’s contract had valued the Belgian at £23.5m.
Everton had already turned down a £12m United bid for Baines earlier in the summer.
The Echo claims Everton believe the bid was made with the intent of unsettling the players.
Everton manager Roberto Martinez has become increasingly bullish about the prospect of keeping his star duo in recent weeks, having openly revealed that one unnamed club – thought to be United – had made a final offer for Baines that had been rejected.
The former Wigan boss had been quoted as saying: "Before we went to Austria one club officially contacted ourselves with an offer and it was a final offer.
"They put something over and the valuation wasn't anywhere near the quality of Leighton Baines.”
Everton’s stance on Fellaini was also heavily strengthened after Martinez confirmed a buy-out clause had expired on July 31.
"Marouane had a buy-out clause where it was very straightforward. If someone met that valuation, the club had no say.
"That finished on July 31 so you can imagine during that period everyone was a bit expectant about what would happen. Once that date went, you just focus on working for the next season.”