Woodman, 20, is highly regarded by the Newcastle hierarchy and has designs on being loaned out to a League One side this season in order to receive regular game-time and further his career.
Woodman only signed a new four-year deal at St James' Park last season, but the Newcastle Chronicle claim that Everton and Crystal Palace are monitoring the former Hartlepool United loanee and could attempt to sign him permanently before presumably sending him to a club residing the third tier of English football.
With Joel Robles and Maarten Stekelenburg still a part of Ronald Koeman's set-up there does not seem to be an immediate place for Woodman, who kept goal as England Under-20s won the World Cup earlier this summer.
Five of Everton's brightest young prospects joined Woodman in helping the young Lions achieve glory in Japan. The Toffees' heavy representation in the tournament was formed by Callum Conolly, Dominic Calvert-Lewin, Kieran Dowell, Jonjoe Kenny and Ademola Lookman, who is believed to be catching the eye of Championship outfit Derby County.
Lookman, 19, joined Everton from Charlton Athletic in January for £11m and featured in 10 matches last season as he attempted to establish himself under Koeman, who described him as the "future" of the club not long after his arrival.
The precocious winger has impressed during his pre-season cameos and is expected to be part of the Everton squad for the current campaign, but Rams boss Gary Rowett is hopeful he can convince the Toffees to temporarily part with the coveted teenager.
Rowett told the Derby Telegraph: "We know there are a lot of names out there, a lot of Premier League players out there that would give us something different, and Lookman would be one of those players."