The Daily Telegraph says the Black Cats would consider selling Pickford, who has three years left on his contract, for a transfer fee of around £25m ($32m), plus add-ons.
Pickford, 23, 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.
Everton, Manchester City and West Ham have also been linked with the 23-year-old.
City's first-choice goalkeeper Claudio Bravo has endured a difficult first season in English football, leading to speculation that manager Pep Guardiola will try to bring in a replacement in the summer.
Meanwhile, United could also be in the market for a goalkeeper if David de Gea ends up moving to Real Madrid.
"At the moment, I really wouldn't know what the odds are on [Pickford] staying," Sunderland manager David Moyes said 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."
Moyes has said he will stay on as Sunderland manager next season despite the club's relegation to the Championship.
"I know what needs to be done to get back in the Premier League," the Scot told the BBC last week.
"I will know more come the end of the season, once we see exactly what we are able to deal with, what we can work with, then we will know exactly what we can do."