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Published 16:59 24 Jul 2016 BST
Updated 17:17 24 Jul 2016 BST
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Phil was part of Moyes' coaching staff at Old Trafford, and worked on his brother's coaching team at Valencia last season.
However, both of the Neville brothers are currently without a club. Gary was sacked as Valencia coach towards the end of last season, and Phil left soon after.
The elder Neville also resigned as England coach following the country's exit to Iceland at Euro 2016.
Paul Bracewell, who acted as Allarydce's assistant last season, and first-team coach Robbie Stockdale, are still at Sunderland, and took charge of the team's friendly against Rotherham on Saturday.
However, there's a chance Moyes may want to bring in his own staff, as he did at United when he succeeded Alex Ferguson.
Steve Round, who was Moyes' assistant at United, is currently a coach with Derby County, so Gary could be in contention for the role.
However, most football fans would like to see him fill the vacancy on Sky Sports' Monday Night Football show.

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