The 24-year-old was a key player for Los Blancos during the 2022-23 campaign, featuring on 56 occasions in all competitions, scoring 12 goals and registering seven assists in the process.
Real Madrid fell short in La Liga and the Champions League, but they won the Copa del Rey, UEFA Super Cup and Club World Cup last season.
Valverde signed a contract extension with the capital giants in August 2021, with his deal running until June 2027, but there has recently been speculation surrounding his future.
A number of Premier League clubs are believed to be interested in the Uruguay international, who made the move to Bernabeu from Penarol in 2016, initially representing Real Madrid Castilla.
According to El Nacional, Liverpool boss Jurgen Klopp is a huge admirer of the South American, and the Reds recently placed a £51m offer on the table for him.
However, the report claims that Real Madrid quickly rejected the bid, with the Spanish giants maintaining that the midfielder is not for sale under any circumstances.
Valverde has turned out for Los Blancos on 204 occasions in all competitions, scoring 18 goals and registering five assists in the process.
A recent report claimed that Real Madrid had branded the midfielder as 'non-transferable', with the club not interested in a sale despite being well-stocked in the middle of the park.
Liverpool have already boosted their midfield this summer with the signing of Alexis Mac Allister from Brighton & Hove Albion, but two more players are expected to arrive in that area of the field.
Both Toni Kroos and Luka Modric are likely to sign new one-year deals at Real Madrid, while Aurelien Tchouameni and Eduardo Camavinga are also midfield options for Los Blancos boss Carlo Ancelotti.
Jude Bellingham's arrival from Borussia Dortmund has also now been confirmed, adding to the competition for a starting role in the middle of the Real Madrid midfield.
Valverde has often been used as a right-sided attacker, though, and he could largely feature in that area of the field next season due to the forward departures at Bernabeu this summer.
Nice midfielder Khephren Thuram is also currently being linked with a move to Liverpool, while Bayern Munich's Ryan Gravenberch is believed to be of interest to the Reds.