The Toffees are on the hunt for a new goalkeeper despite signing Maarten Stekelenburg on Friday
Portugal's Rui Patricio
Krul was linked with a move to Goodison Park following Tim Howard's return to his native United States with Colorado Rapids.
The Dutchman, of whom Toffees boss Ronald Koeman is said to be a fan, has a year to run on his Magpies contract, making him a potential bargain for someone this summer.
Newcastle United's Tim Krul
However, the 28-year-old is currently recovering from a long-term lay-off and his return is not expected before the start of the new campaign, leaving Everton, who added another Dutchman, Maarten Stekelenburg, to their ranks on Friday, likely to look elsewhere.
According to the Chronicle, Sporting's Rui Patricio is one player being looked at by the Toffees, along with Leicester City's Kasper Schmeichel.
Patricio earned his 50th cap for Portugal in Thursday's Euro 2016 quarter-final victory against Poland, in which he saved the decisive spot-kick from Jakub Blaszczykowski.
Rui Patricio saves Jakub Blaszczykowski's penalty
However, a new contract signed in March 2016 could make the 28-year-old difficult to obtain, while Schmeichel may be reluctant to leave Leicester ahead of the Foxes' Champions League debut this coming season.