Everton were quick to reject a joint B£28million bid for the pair last week, branding it "derisory and insulting" on Monday.
However, the Toffees statement lacked the 'not for sale' message that has been attached to all official United responses to Wayne Rooney's current uncertain position at Old Trafford.
And for Fellaini in particular, it seems only a matter of time before Moyes gets his man.
"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," said Everton's new director of communications Alan Myers.
"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 anyway."
Moyes stated earlier in the summer that United do need to bolster their central midfield options.
And whilst United's chase for Cesc Fabregas is yet to be abandoned, realistically, with the former Arsenal skipper starting Barcelona's 7-0 rout of Levante on Sunday, the chances of him exiting the Nou Camp this month are remote.
The pursuit of Baines is more puzzling.
Although the England defender is an obvious talent, and is one of the most dangerous set-piece exponents in the top flight, with his 29th birthday looming in December, the former Wigan defender is evidently a first-team player.
Yet Patrice Evra has recovered to something like his old form after a difficult 2011-12 campaign.
At 32, he is one of United's most experienced players and was named captain in Nemanja Vidic's absence over the past two seasons.
The only time in recent memory the Red Devils had two experienced full-backs vying for the same position was 2006-07, when Evra and Gabriel Heinze went head-to-head.
The rest of the time, a younger player has tended to be the stand-in, so if Baines did eventually join United, it would seem obvious for Evra to move on, even if Moyes has previously insisted that would not happen.
Meanwhile, Moyes has admitted Robin van Persie is even better than he imagined him to be when he first arrived at Old Trafford.
The Dutchman has already scored four goals this season, bagging two more superb efforts in Saturday's win at Swansea after a scoring both goals in United's Community Shield win over Wigan.
"Is he better than I thought? Maybe, yeah," said Moyes.
"His goals on Saturday were top goals.
"I hope he can get better. With his experience he can."