"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," Koeman said on his official Twitter account on Tuesday.
"I would like to thank (chairman) Bill Kenwright, (major 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."
The 54-year-old, who replaced Roberto Martinez at Goodison Park, spent a significant amount of money overhauling the squad in the last transfer window, but Everton made their worst start to a season since 2005-06.
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He is the third Premier League manager to lose his job in the current campaign, after bottom-side Crystal Palace fired Frank De Boer after just five games in charge and Leicester City parted company with Craig Shakespeare last week.
(Writing by Simon Jennings in Bengaluru; editing by Sudipto Ganguly)
