The 28-year-old has been playing through the pain barrier in recent weeks but the decision has been made to have surgery to prevent any long-term problems.
"Leon's ankle problem has been something that he has carried and the way he has performed despite this is a testament to him," said City manager Steven Pressley.
"We obviously need to bear in mind Leon's long term fitness and felt this decision was the right one.
"Hopefully this means he can overcome something that has been troubling him for a while and we can get him back in full health sooner rather than later."
Clarke has scored an impressive 21 goals in 31 appearances this season for Scunthorpe and then the Sky Blues.
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