According to the same story, Pickford is seen as the "perfect" replacement for De Gea should the Spaniard indeed leave the Red Devils.
De Gea faces competition for his global place from goalkeepers including Chelsea's Kepa Arrizabalaga, and Saul came out in defence of the United star ahead of a friendly against Bosnia-Herzegovina on Sunday.
Neville, speaking ahead of England's Nations League victory over Croatia, revealed he had serious doubts about Pickford but the former United defender is coming round to the 24-year-old.
Pickford became the most expensive British goalkeeper in history when completing a £25m ($32m) switch from Sunderland to Everton in the summer of 2017.
De Gea made a single save at the World Cup, where he was guilty of a poor mistake that allowed Cristiano Ronaldo to score in a draw against Portugal, and his worldwide form has not improved since.
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"If David concedes goals, I do not think it's just his fault, but everyone's, we all have to pay more attention".
"I actually think worldwide football suits him a lot in the sense of the way the team play".
"I think the fact you get more time to play out a the back than you normally would in the Premier League".
'I think he's suited to worldwide football. The England number one's rise to the top has been rapid in recent months and he was probably England's best player in the World Cup this summer. I always though a keeper should be dominant, ' he added.
"I was always of the Peter Schmeichel, the big presence".
'He doesn't look huge in the goal.