The Algemeen Dagblad newspaper said Liverpool made contact with Feyenoord on Wednesday morning.
Slot was heavily linked with a possible move to Tottenham Hotspur last year, but opted to stay at Feyenoord and signed a contract extension with the Dutch side.
He has been widely praised for his innovative, attacking football with Feyenoord, winning the league title in the 2022-23 season, as well as the KNVB Cup on Sunday.
In his first season at Feyenoord, he also led the Rotterdam club to the final of the inaugural Europa Conference League.
Reuters have contacted Liverpool for comment.
Klopp announced in January he would stand down as Liverpool manager at the end of the season, after a successful spell at the club in which he won the Champions League, Premier League, Club World Cup, FA Cup, League Cup and Super Cup, as well as the Community Shield.
In his final campaign, Liverpool have won the League Cup and remain in contention for the Premier League title.
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