Clingan had to pull out of the Northern Ireland squad for Wednesday's friendly with Cyprus having come off second best in a challenge from St Mirren defender Sean Kelly last weekend.
He joins full-back Sean Clohessy (ankle) and Darren Barr (hip) on the absentee list in the Rugby Park treatment room.
Kilmarnock lost 2-0 in Paisley to extend their winless sequence to five games.
Hearts manager Gary Locke is demanding a response from his players after last week's 4-1 defeat at Motherwell.
He said: "I kind of knew we would probably have a wee dip again with the team being so young, but it was a bad performance and there are no excuses. If we've only got two players who turn up, you don't have much chance of winning.
"Hopefully it's only a blip and we can get back to the performances we were putting in before Motherwell. The boys had been doing well and we were all disappointed with what happened."
Callum Tapping remains doubtful because of a hamstring niggle while fellow midfielder David Smith (knee) and striker Jamie Walker (foot) remain long-term absentees for the Premiership's bottom club.
But Jordan McGhee, Callum Paterson, Gary Oliver and Sam Nicholson are all available after they came through their midweek Scotland youth fixtures unscathed.