The Telegraph says the Magpies are currently preparing an offer of 15m for the Arsenal goalkeeper.
If an agreement is reached, the 26-year-old will receive a five-year contract.
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At the Gunners, Ramsdale lost first-choice status to David Raya earlier this season.
The potential arrival of Ramsdale would mean the departure of Martin Dubravka. The Slovakian would not be satisfied with the position as number three behind Nick Pope (32) and Ramsdale.