Killie owner Billy Bowie has been inundated with applications following Clarke's departure to Scotland — and will begin the interview process this week when he returns from holiday.
The Uruguayan is among a number of candidates the Rugby Park club are considering for their vacant manager's post.
Poyet has links to Kilmarnock after arranging for his son, Diego, to spend a couple of months training with the Ayrshire outfit last season when he was between clubs.
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