Talks with Gueye are said to be progressing well, while Coleman has been offered a new deal, with manager David Moyes particularly keen to retain the club captain in a playing role.
The Toffees entered the summer with over a dozen first-team players approaching the end of their contracts.
Most of those situations have now been resolved, though negotiations continue with Dominic Calvert-Lewin and Michael Keane over potential extensions.
Several long-serving and senior figures are confirmed to be leaving. Mason Holgate will depart at the end of the month, along with Joao Virginia, Abdoulaye Doucoure, Asmir Begovic, and Ashley Young.
Neal Maupay, meanwhile, will remain at Marseille permanently after his loan deal was confirmed to include an obligation to buy.
On the loan front, Armando Broja, Jesper Lindstrom, Orel Mangala, and Jack Harrison have all returned to their parent clubs following the conclusion of their temporary spells at Goodison Park.
However, Everton have secured the first signing of the window by turning Carlos Alcaraz's loan into a permanent deal.
Everton are making waves in the transfer market by reportedly targeting Manchester City's Jack Grealish, aiming to secure a high-profile signing to kickstart a new era under their ambitious ownership.
Everton appear to be interested in Villarreal striker Thierno Barry, as manager David Moyes looks to strengthen his attacking options ahead of the new season.
Jarrad Branthwaite's stock continues to rise after an outstanding season at the heart of Everton's defence, with Chelsea showing keen interest.
Everton have confirmed they will complete the permanent signing of Carlos Alcaraz at the conclusion of his current loan spell, with the midfielder set to remain at Goodison Park until at least June 2027.
