It claims defender Lamine Kone - who only joined Sunderland in January - has travelled to Merseyside ahead of an £18 million switch to Everton and Moyes wants the veteran centre-half to replace him.
The report does not state whether it could be a player-plus-cash deal or simply two separate transfers.
The 27-year-old is hugely popular on Wearside after an excellent first five months with the Black Cats, and scored twice in the 3-0 win over Everton which secured Sunderland's Premier League status - and relegated bitter North-East rivals Newcastle United in the process.
As a result of his popularity, Sunderland fans seem angry not only at the prospect of losing him, but the idea that the Everton man - who turns 34 this month - is his direct replacement.
Jagielka in for kone is like replacing phil taylor in ur darts team with stevie wonder.— Frank Drebin (@FrankDrebin8) August 9, 2016
#SAFC Jagielka replacing Kone would be like trading in a five star hotel or a two star one— Daniel Jenks (@DanielJenks89) August 9, 2016
If Kone leaving and us signing Naismith and Jagielka is the culmination of #SAFC's transfer dealings this week, then I'm done.— Michael Bartley (@schmeichel1b) August 9, 2016
@SAFCBuffalo Better spend all this money well and reinforce. Hope Jagielka can hold up for half season and buy a young CB in Jan.— Matthew M Rice (@mam24rice) August 9, 2016
#SAFC Jagielka and O'Shea would be a great defensive partnership ... five years ago— Daniel Jenks (@DanielJenks89) August 9, 2016
@Wadders30 you seen Moyes wants Jagielka for Laminè Konè? Madness mate— Con™ (@ConMcCoy11) August 9, 2016
If Sunderland sell @LamineKone_23 and replace with Phil Jagielka David Moyes may as well resign now to save him getting sacked in December— Morgan Lowrie (@MorganLowrie) August 9, 2016
Moyes brought the England international to Goodison Park from Sheffield United in 2007 and has been one of Everton's stand-out performers for over a decade.
But Sunderland fans' reception of the speculation is hardly surprising given his age, and of course how influential Kone is at the Stadium of Light.