The Ibrox club rejected a £5million bid in the summer for their 20-year-old star.
Manager Steven Gerrard labelled the bid a joke.
He said at the time: “I’ll ask the question when I go back on the bus but if you’re going to talk about Nathan Patterson you need to get real.
"£5m is, I don’t know, that’s definitely come out of a joke book surely.”
Now Everton, according to the Daily Mail, are preparing to come back with an increased offer.
They were reported to be returning with a £10million bid in the summer before turning their attention to a loan deal for Arsenal’s Ashley Maitland-Niles.
Patterson, who has started four games for Rangers this season, is highly-rated at Ibrox and has attracted plenty of admiring glances with Rafa Benitez understood to be a big fan.