The 23-year-old is regarded as one of England's most promising keepers and was one of the few successes in a Sunderland side who finished bottom of the Premier League last season.
Echo said the deal, which includes various add-ons could eventually rise to 30 million pounds, just short of the world-record 35 million pounds which Manchester City recently paid Benfica for Brazilian Ederson Moraes.
Pickford is currently in Poland with the England under-21 team and the move will not be completed until July 1 when the international transfer window opens.
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Everton are expected to be one of the most active teams in England's transfer window and British media have also reported they are in discussions to buy Malaga striker Sandro Ramirez and Ajax midfielder Davy Klassen.
(Reporting by Neil Robinson; editing by Sudipto Ganguly)
