After securing their Premier League survival on the final day of the season, Everton have started their preparations for the 2023-24 campaign.
The Toffees have confirmed that they have offered extensions to Coleman and Davies, who are both out of contract at the end of the month.
Coleman, who joined Everton from Sligo Rovers in 2009, is widely expected to put pen to paper on a new deal.
The Republic of Ireland international has made 409 appearances for the Toffees, chipping in with 28 goals and 32 assists.
According to The Guardian, Davies is considering options before he decides whether to extend his stay with the club.
The 24-year-old made just 19 Premier League appearances in the 2022-23 campaign, and he is yet to start a game under Sean Dyche, who was appointed Everton boss in January.
Davies, who has spent the entirety of his career with Everton, is believed to be attracting interest from a number of Premier League clubs.
Everton have also offered a contract extension to third-choice goalkeeper Andy Lonergan, but his fellow shot-stopper, Asmir Begovic, will leave the Merseyside club after rejecting a new contract.
The 35-year-old will be followed through the exit door by Andros Townsend and Yerry Mina after Everton decided against offering the pair fresh terms.
Everton academy product Isaac Price will also depart after turning down a new contract, with the 19-year-old midfielder set to join Belgian side Standard Liege.
The Toffees recently confirmed that they had activated the option to extend Abdoulaye Doucoure's contract until the summer of 2024.
Doucoure scored the winning goal in Everton's final-day victory over Bournemouth to keep his side in the Premier League.