The Toffees are said to have had a three-man shortlist of potential additions as they entered the January transfer window.
And the availability of Spaniard Adrian at the London Stadium had drawn their attention.
The West Ham man has lost his starting role of late to Darren Randolph in east London, with Manchester City's Joe Hart set to be pursued at the end of the season.
Adrian has been at West Ham since arriving from Real Betis in 2013
Despite discussing the possibility of signing 30-year-old Adrian, Ronald Koeman opted to hold off
However, according to the Liverpool Echo, despite discussing the possibility of signing 30-year-old Adrian, Ronald Koeman opted to hold off.
It is not known if Everton will follow up their interest later this year in the Hammers keeper, who has history with the Toffees having scored a winner against them in an FA Cup third round replay penalty shootout.
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But a new long-term No 1 is still on their agenda.
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Joel Robles' form has impressed Koeman since coming back into the team for summer signing Maarten Stekelenburg.
And Everton will now stick with the two until the end of the season.