Clarke was appointed Scotland head coach 24 hours after guiding Killie to the Europa League with a 2-1 victory over Rangers.
A third-place finish – Killie’s highest since 1966 – secured a return to Europe after an 18-year absence and saw them break their record top-flight points tally for a second season running.
Clarke had claimed his major success was attracting three stands full of Killie fans to watch them host Rangers and the club have expressed their gratitude at his work over 19 months in Ayrshire.
A statement read: “Steve departs Rugby Park as a Killie legend, having twice broken our record points tally and leading the club to our highest league finish in 53 years and a place in the Europa League qualifiers.
“Since joining in October 2017, Steve and his assistant Alex Dyer have led the team to back-to-back top-six finishes but more importantly have helped restore a feeling of pride, passion and belief in Scotland’s oldest professional football club.”
Majority shareholder Billy Bowie added: “On behalf of the board of directors and everyone connected with Kilmarnock FC, I’d like to express my sincere gratitude to Steve for his passion and dedication to job.
“While we’re naturally disappointed to lose such a talented manager, I understand the lure of managing Scotland is a powerful one.
“Steve leaves an incredible legacy, delivering our best campaign in over half a century and providing European football for the first time since 2001. His place in this club’s illustrious history is assured and he will always be welcomed back to Rugby Park with open arms.”
Clarke, whose brother Paul was a long-serving Kilmarnock player, had stated his aim was to give the fans and town some cheer when he took over.
He said in a statement: “It has truly been an honour and privilege to work for Kilmarnock Football Club and I will always treasure the moments and memories created here.
“I’m extremely proud of all our achievements during my time in charge and I’m especially pleased that I was able to play a part in putting a smile back on the faces of Killie supporters.
“A manager is nothing without a group of talented players and staff, willing to work hard and listen to instructions so I’d like to thank them for their support during my time at Rugby Park.
“I’d like to express my gratitude to the board of directors for all their backing and ask you to support them as they continue to build this fantastic football club.”
Clarke’s assistant, Alex Dyer, and Livingston boss Gary Holt, a Scottish Cup winner with Killie, were among the early favourites to succeed the former Chelsea defender with the bookies.
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