According to reports, Manchester United launched a double bid for the aforementioned stars before this weekend's opening day Premier League action.
Everton subsequently rejected the bid as it was deemed nowhere near their valuation for the highly-rated duo before releasing a statement on their official website lamenting Manchester United's offer.
Manchester United's bid comes just a few weeks after Fellaini's £22million release clause expired - which makes the joint bid even more surprising.
New Red Devils manager David Moyes has been linked with a move for Baines - in particular - throughout this summer's transfer window.
In fact, Moyes made a £12million bid for England left-back Baines in July - which was also flatly rejected.
Everton boss Roberto Martinez said last week that he was not 'concerned' about Baines and Fellaini leaving Goodison Park.
With two weeks remaining of the transfer window, it is likely Moyes will make a renewed bid for Fellaini - as a midfielder is said to be high on the 50-year-old's agenda.