Wigan Athletic’s Roberto Martinez is heavily fancied to leave the relegated club and succeed Moyes, who met with his new Manchester United squad on Monday.
However, should he not be in a position to take the job it is anticipated O’Neill, 61, could make a quick return to the Premier League after being sacked by Sunderland in March.
Last night he was also being linked with Stoke City after Tony Pulis left the club by mutual consent after seven years in charge. But it is understood the Toffees are keen on O’Neill.