The 27-year-old Costa Rican has been enjoying a fine season between the sticks for the Spanish side, helping them move into a solid seventh position as it stands.
His excellent displays have alerted a host of clubs, with the Merseyside rivals believed to be two that are particularly keen.
Navas is set to represent his nation at the World Cup – where they will take on England – and after the tournament he will be in the final 12 months of his contract.
Levante are likely to sell at that point if they cannot extend his stay, with gaining a fee for his signature far more tempting then keeping him for a year before losing him for free.
Both Liverpool and Everton are said to be looking to snap up new back-up ‘keepers, with Simon Mignolet and Tim Howard short on competition at Anfield and Goodison Park respectively.
Navas is also wanted by Sevilla and Atletico Madrid.
The stopper’s compatriot and former Manchester City, Derby and West Ham forward Paulo Wanchope has previously stated his belief that he could be a star in the Premier League:
“Keylor would be a hit in England because of his quality.”