Hazard was the matchwinner on Saturday as the 'Blues' defeat Manchester United in the FA Cup final, but their fifth-placed finish in the Premier League and the continuing rumours over the future of manager Antonio Conte mean that they may be forced in a summer rebuild.
The former Lille man is without a doubt the jewel in their crown and with Real Madrid believed to hold a long-standing interest in the Belgian, Clarke believes it may be time to cash in.
"You can always look to replace players," the Kilmarnock told 'Sportsweek' on 'BBC Radio 5 Live'.
"They come and go, they always move on. The club is the most important thing.
"They have to rebuild that squad a little bit. It wasn't good enough to be in the top four. They have got a lot of work to do over the summer.
"To help fund other transfers for the club this summer maybe they do have to sell Hazard."