Both players could leave Goodison Park when their contracts run out at the end of this month, but Everton are keen to hang on to them after surviving relegation this season.
It is the second time the Toffees survived the drop by the slimmest of margins, and they are keen to avoid a third crack at flirting with the prospect of second-tier football.
Coleman missed the final four games of the campaign with a knee injury that required surgery, but the 34-year-old remains an influential figure in the dressing room.
The Irishman has been associated with the Merseyside club over the last 14 years since joining from Sligo Rovers in 2009, and they want to rely on his invaluable leadership for at least another campaign.
Coleman is expected to extend his stay at Goodison Park and to be fit for the start of next season, but Everton could have trouble convincing Davies to commit his long-term future to the club.
Davies came up through the youth ranks at Everton but has endured an injury-plagued campaign, making just 20 appearances across all competitions.
He has only managed substitute appearances since Sean Dyche replaced Frank Lampard in the dugout and is reportedly attracting interest from other Premier League clubs.
Meanwhile, Asmir Begovic, Andros Townsend and Yerry Mina are expected to leave Goodison Park this month.
Begovic turned down a contract proposal from the club, while Townsend has not featured for the Toffees since suffering an anterior cruciate ligament knee injury in March last year.
Mina was one of the more consistent performers in Dyche’s side toward the end of the season, putting in some stellar defensive displays that helped his side avoid the drop.
But he is also expected to leave as a free agent.
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