The 25-year-old impressed for Uruguay on their run to the quarter-finals of the 2018 World Cup.
Laxalt has a contract with Genoa until 2022, and entered this summer's World Cup off the back of scoring three times and registering three assists in 32 Serie A appearances last term.
According to The Sun, both Everton and Bournemouth were impressed by Laxalt's performances in Russia, and the Premier League duo are preparing £18m offers in a bid to secure his services.
That said, the same report claims that Crystal Palace are also interested in signing the versatile left-sided player.
Laxalt came through the youth system at Uruguayan club Defensor Sporting before joining Inter Milan in 2013, although he did not make a single appearance for the Italian giants ahead of a move to Genoa.