The two sides are said to be keen on the services of Bernat, who has been capped for Spain at several youth levels.
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Valencia's well-documented financial problems have seen them sell many of their best players in recent years, with David Silva, Juan Mata, David Villa and Roberto Soldado all leaving the club for big-money fees.
But the club's sporting director, Francisco Rufete, insists Valencia are determined to keep Bernat - who can also play on the left side of midfield - in Spain, despite foreign interest.
"Juan is a home-grown player from our academy and is a very important player for us," Rufete is quoted as saying in the Italian media.
"We want him to stay with us for a long time, but we also know that there are many clubs that are interested in him because he is playing at such a good level.
"It is good that the team continues to produce such results and Bernat is certainly a player that interests Napoli and many other clubs, but our intention is to keep him."