Davies, who is a product of the Toffees academy, has made 175 appearances for the club, but this season he has been playing a bit part role for the club.
Doucoure joined Everton from Watford in the summer of 2020 and was a regular starter for the Toffees.
Earlier this season, the Mali international lost his place under Frank Lampard, but reclaimed his spot in the starting line-up after the arrival of Sean Dyche.
Davies and Doucoure both have contracts with Everton that expire at the end of this season and will be free agents in the summer.
It has been claimed that Everton are reluctant to make decisions over the futures of the two midfielders at Goodison Park due to the uncertainty over their Premier League survival.
If Everton are relegated to the Championship, they may want to trim their wage bill and may opt against offering new deals to the pair.
Everton boosted their survival hopes on Saturday when they edged out Brentford 1-0 at Goodison Park.