The 23-year-old Georgian goalkeeper, whose move is subject to a work permit and international clearance, will spend the rest of the current season with his team before moving to Liverpool next summer.
Mamardashvili made 102 appearances for Valencia.
British media reported that the deal was worth about 30 million pounds ($39.77 million).
Mamardashvili enjoyed an impressive run at the European Championship with Georgia, who reached the last 16 in their first appearance.
His 21 saves in the three group stage matches were the most in the history of the continental tournament.
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