Pickford, who made the switch to the Toffees last summer from Sunderland, enjoyed a brilliant debut season at Goodison Park and his impressive performances earned him the number one jersey for England.
The 24-year-old, who was an ever-present for Everton last term, produced some eye-catching displays for the Three Lions as they reached the semi-finals of the 2018 World Cup in Russia.
Reports are already linking the shot-stopper with a move away and Seaman, who is regarded as one of England's greatest keepers, believes Pickford is likely to move to a bigger club.
He told the Daily Star: "Jordan Pickford will probably end up at one of the top six clubs in the Premier League if he keeps progressing. He'll make it hard for the big clubs to ignore him.
"He needs that challenge, because he's not even played Champions League football yet. He's just skipped that and gone straight to the World Cup and showed he can do it there. He had a great tournament.
"He's a quality goalkeeper and has got to keep doing it."