The announcement comes exactly a year after the 44-year-old was appointed at Rugby Park, leaving the club sixth in the table.
Clark steered Killie to safety last season through a 4-1 aggregate relegation-promotion playoff victory over Falkirk, ensuring a 24th consecutive season in the Scottish top flight for his now-former side.
"I have had a wonderful year as Kilmarnock manager. The club and fans have treated myself and my family fantastically - we are in a good position with a strong and loyal squad and staff," he told the official Shakers site.
"The directors have given me their full support in rebuilding the squad and the fans will always have a special place in my heart but I still have huge ambition to one day manage at the highest level in England, which is obviously the Premier League, and I feel that the fresh challenge of managing a club in England's League One could provide me with the platform to build towards this goal."