The London Evening Standard says the pair are the top two candidates to succeed Allardyce should be forced out.
Pulis, who has reached a settlement with Palace, left Selhurst Park as a consequence of irresolvable differences over transfer targets this summer, with the former manager unwilling to accept that Parish was doing all he could to sign the players who had been identified.
Moyes, who spent 11 years at Everton, has been without a club since he was fired by Manchester United less than a season into the role. The Scot is believed to have been interviewed by Galatasaray since then, but is yet to return to football management.
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