According to Calcio Mercato, the 25-year-old is targeted by a whole host of clubs from all across Europe - not least Marseille, Benfica and Valencia. Three Bundesliga clubs are also believed to have entered the race for the Uruguayan - Schalke, Wolfsburg and Stuttgart.
Several of Genoa's Serie A rivals have also been linked with Laxalt, including Fiorentina, AS Roma and Lazio.
Laxalt enjoyed an impressive World Cup with Uruguay, for whom he played as a left wing back. His audacious shot took a wicked deflection for La Celeste's second goal in their 3-0 group stage win over hosts Russia.
The Eagles may struggle to compete with some of their rivals for Laxalt's signature - not least Roma, who can offer the Uruguayan Champions League football. However, Genoa would reportedly be prepared to sell him for €20m, which would be well within a Premier League club's price range.
Palace are far from being the first English side to show an interest in Laxalt. He has previously been linked with moves to Liverpool and Everton, among other clubs. The Eagles' interest in a player of Laxalt's calibre reflects their seriousness about building for a long-term future in the Premier League.
Palace came worryingly close to relegation last season, making a catastrophic start before former England manager Roy Hodgson steadied the ship and guided them to 11th place in the Premier League table. With Laxalt on board, they would have an exceptionally versatile player who can play either as a full back or as an attacking midfielder.
Laxalt has played 68 Serie A games for Genoa, scoring four goals. He has also featured ten times for his country, though he is yet to score at international level.