The 28-year-old has been playing with the problem for the past few weeks as the Sky Blues battle for promotion from League One.
"Leon's ankle problem has been something that he has carried and the way he has performed despite this is a testament to him," Coventry manager Steven Pressley is quoted as saying by Sky Sports News.
"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 10 goals in 16 games for Coventry since his January move from Scunthorpe United.