United lodged a bid earlier this week believed to be in the region of £28 million which was turned down by Everton, who labelled the offer as "derisory and insulting".
Everton boss Roberto Martinez has since insisted that no further offer have been made for the pair, but Goal report that the Red Devils have increased their bid to £35 million.
According to Goal the bid breaks down as £20 million for Fellaini and £15 million for Baines.
Everton are likely to value Fellaini at much higher than that having paid around £15 million for his services in 2008, however Goal report that Manchester United won't go any higher with their offer for the 25-year-old.
Manchester United boss David Moyes has made signing his two former players a priority in the final stages of the transfer window having already missed out on two big-name targets already.
He failed with a move for Thiago Alcantara who eventually joined Bayern Munich, while two bids with his Barcelona's Cesc Fabregas were also unsuccessful.
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