Clarke says Rodgers faces the same fate as Kenny Dalglish next spring unless he steers them into the Champions League places.
Clarke insisted: "The owners have said top four is not imperative for the current season. I don't believe that.
"A club like Liverpool have to be there challenging for the top four - I don't think they can accept anything less.
"But we stabilised the club, got them in a good position, reached two cup finals and put together a decent squad.
"I spoke to Kenny towards the end of last season, when we didn't envisage we would not be at Liverpool.
"We spoke about our prospects for this coming season and both Kenny and I had agreed top four was the realistic target for that squad. Kenny and I were pretty sure with the group of players we had, Liverpool will be in the top four this season."