The Reds are believed to have expressed an interest in the Uruguay international during the recent transfer window before switching their attention to other targets.
The 25-year-old made the move to Bernabeu from Penarol in 2017, and he has represented Los Blancos on 209 occasions in all competitions, scoring 19 goals and registering 15 assists.
Valverde's most productive season on a personal level came in 2022-23, when he scored 12 goals and registered seven assists in 56 appearances in all competitions.
The South American has a contract with Carlo Ancelotti's side until the summer of 2027, and he is viewed as one of the team's most important players.
However, there has been speculation surrounding his future, with Liverpool believed to view him as a leading target in the middle of the park.
Chelsea and Manchester City are also said to be admirers, and Valverde has said that he could seek a move away from Real Madrid at some stage in the future.
"Premier League offers? I have the possibility of playing for Real. The day I don't feel like that, I will probably look for another solution or Real itself will look for one for me," Valverde told reporters.
The midfielder has again been an important player for Ancelotti's side at the start of this season, featuring in all five of their La Liga matches, starting four of them.
Valverde has won two La Liga titles, one Copa del Rey crown, two Spanish Super Cups, the Champions League, the UEFA Super Cup and two Club World Cups during his time at Madrid.
The midfielder was on the scoresheet against Real Sociedad at the weekend, and his performance led to teammate Joselu branding him "an animal".
"He's a beast, an animal, you see him running and it seems that his strength will never run out," Joselu told reporters.
Real Madrid have won all five of their La Liga matches at the start of the new season, and they will now be aiming to start their Champions League group-stage campaign with a win, welcoming Union Berlin to Bernabeu in the competition on Wednesday night.