Clarke, chief operating officer of the Walmart-owned supermarket, is now firm favourite to take the position according to reports, following the resignation of its trading director, Darren Blackhurst.
Last month, Asda's chief financial officer, Judith McKenna, and David Cheesewright, who runs Wal-Mart Canada, ruled themselves out of the running to replace Bond.
Bond announced that he was stepping down last month after five years in the role, but will become chairman of Asda's executive committee.
An Asda spokeswoman declined to comment.
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