The Times says Newcastle manager Eddie Howe is a fan of the 23-year-old Georgian's play. The keeper has been repeatedly watched by Toon scouts this season.
However, it is expected that Newcastle will face competition in the fight for Mamardashvili.
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The goalkeeper has been playing for Valencia since 2021.
This season he's played 27 matches for the team in La Liga, conceded 31 goals and kept 9 clean sheets.