Manchester United look certain to return with a higher offer after Everton branded their £28million joint bid for Leighton Baines and Marouane Fellaini as "derisory and insulting".
But it is clear he will need to offer substantially more than the respective £12million and £16million for Baines and Fellaini which is believed to have been the split for the offer, which was lodged at the end of last week.
"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 director of communications Alan Myers.
That both players started Saturday's 2-2 draw with Norwich underlines the fact neither is in dispute with the Toffees.
However, it could easily be argued that the duo would welcome the opportunity of a move to the reigning champions and, having so far failed to reinforce his squad, it would be a major surprise if Moyes backed away so quickly.
Indeed, it could be argued the offer was only in line with the first £20million bid made by Chelsea for Wayne Rooney in the middle of last month which the Blues are tipped to increase still further with a third bid before turning their attentions elsewhere.
A further move for Fellaini could be anticipated on the strength of Moyes' assessment earlier in the summer that United do need to bolster their central midfield options.
United's chase for Cesc Fabregas is yet to be abandoned.
Fellaini might not have shown Fabregas' game-changing abilities. At Premier League level though, the Belgian is an effective performer, and with a date for Darren Fletcher's return to fitness as yet unknown, Moyes currently only has three central midfielders, plus 39-year-old Ryan Giggs, to choose from.
The pursuit of Baines is possibly a more puzzling one.
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.