United manager David Moyes is very familiar with England left-back Baines and Belgium midfielder Fellaini from his time at the helm of the Goodison Park side.
And it has long been speculated that he might be keen to take one or both of the duo with him to Old Trafford.
However, Everton are determined to retain two of their most important players and released a club statement which read: "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."
Baines and Fellaini were both in the Everton line-up on Saturday as the Merseyside club began Premier League life under new boss Roberto Martinez with a 2-2 draw at Norwich.
Fellaini was signed by Moyes for Everton in 2008 for £15m, which rose to £17.5m with add-ons.
But it could be argued United are only offering around £16m for the player, considering they have previously had a £12m bid for Baines rejected.
According to reports, Fellaini's contract included a £23.5m release clause, but that expired at the end of July.
Martinez revealed last week: "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.
"I'm not concerned at all at the moment (about players leaving).
"The window is open and until it closes you're going to get speculation, you're going to get a lot of talk, not just at Everton. It's the norm."
And the Everton boss also confirmed earlier this month that they had received an offer for Baines, but insisted it fell well below their valuation.
He told the Liverpool Echo: "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.
"That was the end of it but since then, there's been all the speculation, which has been part of what the transfer window is.
"There's never been another club or anyone else putting in an offer for Leighton. Since then, we've moved on and we're continuing to focus on the start of the season."