The France international has attracted widespread interest from across Europe since joining the Reds from RB Leipzig in 2021.
He has scored three goals and registered one assist in 29 appearances across all competitions.
The 23-year-old only made 11 top flight appearances as injuries have restricted him to a mere four matches this campaign.
Nevertheless, the France international was included in the 2022 World Cup squad and made five appearances at the tournament, including a semi-final win over Morocco.
Konate is contracted to Liverpool until the end of the 2025-26 season, but according to Mediafoot, Los Blancos are considering a move for the centre-back this summer.
The report adds that Real Madrid boss Carlo Ancelotti has informed the board of his desire to bring Konate to the Bernabeu for the 2023-24 season.
Liverpool currently sit sixth in the Premier League table with 28 points.