After spending more than £100m on new players in the summer, the Toffees failed to live up to expectations on the pitch, winning just two of their nine Premier League games.
The club, who finished seventh last season, are also on the brink of Europa League elimination having lost two and drawn one in the group stage.
Koeman has now broken his silence, writing on Twitter: "I would like to place on record my thanks to the players and staff for all of their work and commitment during my 16 months as Everton manager.
"I would like to thank (chairman) Bill Kenwright and (majority shareholder) Farhad Moshiri and the Everton board for the opportunity to have managed a great club, and to the fans as well for their passionate support for the club.
"Naturally, I am disappointed at this moment, but I wish the team good luck in the future."