Ronald Koeman has been looking to bring in a first-choice goalkeeper since the summer, but will now wait until the end of the season in order to try and strengthen between the sticks.
Adrian has fallen down the pecking order and has not started a league game for West Ham since 5 November. Everton were alerted to Adrian's availability and held preliminary talks with the Irons, but The Liverpool Echo claims Ronald Koeman has been impressed with Joel Robles, who has replaced Maarten Stekelenburg in recent weeks.
Robles has kept four clean sheets in five Premier League starts and even provided an assist for the Blues against Leicester City last month.
The former Wigan Athletic goalkeeper's form has made such an impact on Koeman that he saw fit to rip up his three-man list of goalkeeping targets and decide against signing Adrian, who is out of contract at the end of the season.
The Dutchman and Everton's director of football, Steve Walsh, now have extra time to assess the situation, while Robles could now stake a claim to be the Toffees' long-term number one. The Spaniard will keep his place for his side's trip to face Stoke City at the Bet 365 Stadium on Wednesday (1 February) night.
Everton bolstered their first team with the big money signings of Morgan Schneiderlin and Ademola Lookman but the Goodison Park hierarchy have mainly used the January transfer window to rid themselves of their unwanted fringe players.
Gerard Deulofeu, Oumar Niasse, Darron Gibson, Bryan Oviedo and Tom Cleverley have all left Merseyside this month, while misfiring forward Arouna Kone is set to join Crystal Palace on a deal that will run until the end of the season.