Parker has been in charge as caretaker since Claudio Ranieri's sacking six weeks ago, and it has not been ruled out giving the former England midfielder the job full time this summer, with the club already relegated and facing Championship football next season.
Clarke's name has been discussed at boardroom level as Fulham plot what to do next season, with his experience in the top-flight and Championship regarded as potentially vital if Fulham are to bounce straight back.
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