The Sunday Mirror says the Black Cats want more than double that amount to consider selling the 23-year-old, who has also been linked with Arsenal, Liverpool, Manchester City and Manchester United.
The paper adds that Everton are likely to return with an improved offer.
The Toffees are also interested in Leicester City goalkeeper Kasper Schmeichel, but a move for the Denmark international is complicated by the fact that he does not have a buyout clause in his contract.
Leicester are likely to demand more than £30m for Schmeichel, who has four years left on his contract.
Pickford has won plaudits for his performances after he took over from Vito Mannone as Sunderland's first-choice goalkeeper in August.
He was nominated for the PFA Young Player of the Year award but lost out on the prize to Tottenham midfielder Dele Alli.
Sunderland's relegation to the Championship has raised the prospect of Pickford leaving the Stadium of Light to continue playing Premier League football.
"At the moment, I really wouldn't know what the odds are on [Pickford] staying," Sunderland manager David Moyes was quoted as saying by the Daily Telegraph after his team's 2-0 win at Hull City on 6 May.
"Obviously, it's up to the club, but if Jordan was to go elsewhere and not be the number one, that wouldn't be good for him.
"You can see what he does for us with the saves he makes. He's been outstanding from day one. The saves he made against Hull, they weren't even special, he makes them all the time."
Pickford was named the Sunderland Supporters Association's Young Player of the Year last week, with Jermain Defoe voted the club's Player of the Year.
"I go into every game hoping to keep a clean sheet, if I do that I am happy," Pickford told the Sunderland Echo after accepting the award.
"The Premier League is the best league in the world, all I can do is do my best and if I keep a clean sheet then I am happy."