The Lilywhites are bracing themselves for an offer from the Black Cats for their current boss, who has guided the club from League One and into the top half of the Championship during his four years at the helm.
It is claimed by the Daily Mail that Preston chiefs have already put contingency plans in place, drawing up a shortlist of candidates to potentially succeed Grayson at the Deepdale helm.
Among those included is Moyes, 15 years after he left North End for a successful stint at Everton, which would end on a sour note as he then departed for Manchester United and saw his career deteriorate.
Moyes also failed to impress in spells with Real Sociedad and Sunderland, departing the latter this summer after their relegation to the Championship, but Preston are said to be willing to give the Scotsman a chance.
Grayson claimed last week that he is "fully committed" to the Lancashire club, however, and is preparing for the 2017-18 campaign.