The United manager has upped his offer for the pair to £38m after his initial bid was knocked back earlier last week.
Everton described United's first bid as "insulting and derisory", but Moyes has improved on that to the tune of another £8m.
The report claims Everton are furious with Moyes for returning to Goodison Park in an attempt to sign two of their top first-team players, but the Toffees are yet to officially turn down this second approach.
In fact, Everton are reportedly willing to do business with United - but only for Fellaini.
They want to know how much the Premier League champions are willing to pay for the Belgium international alone, with Everton believed to be open to accepting a standalone £25m bid for Fellaini.
And, if United are willing to negotiate a deal, a transfer could be completed before the weekend with Everton reportedly eager to snap up a replacement as soon as possible. They have their eye on Wigan's James McCarthy but face competition from Newcastle and Tottenham.
A deal could ultimately depend on Wigan's insistence on Everton meeting their £15m valuation on McCarthy - Everton value the Ireland international closer to £12m.
But with the transfer window closing next Monday, there's not much room for manoeuvre for either Everton or United.
Moyes is under pressure to strengthen his squad before the deadline, with central midfield thought to be the main area of concern for the Old Trafford boss.