Davies has impressed since breaking into Everton’s first-team towards the back end of last season.
The 18-year-old midfielder has been handed five Premier League substitute appearances, and he started in Everton’s most recent match against Southampton.
Davies’ performance against the Saints further confirmed the potential that he has, and he capped off a fine display by claiming an assist in Everton’s 3-0 win over Claude Puel’s men.
Davies could now retain his place in Everton’s starting line-up, with Koeman’s options in central midfield limited.
James McCarthy is currently out injured, while Idrissa Gueye is set to play at the African Cup of Nations.
And Koeman said: “He was good, strong in the midfield. He was comfortable on the ball, one or two mistakes but that was more in the first half.
“He was really good and we know he is developing really well at this level.
“We need the boy because we will lose Gana to the Africa Cup of Nations and James McCarthy is still not fit. We don’t have a [large] number of players for that position.”
Koeman has shown great faith in some of Everton’s younger players this season, with Mason Holgate and Dominic Calvert Lewin also getting fairly regular action in the first-team.
The Toffees are currently in seventh place in the Premier League table.