Robertson, 32, confirmed earlier this month that he would leave Liverpool as a free agent when his contract expires in June, bringing to an end a nine-year association with the club.
Fabrizio Romano confirmed this week that Robertson had agreed personal terms with Tottenham — but also flagged that the deal was conditional on Spurs maintaining their Premier League status. Tottenham currently sit 18th in the league, two points from safety.
Atletico Madrid had already attempted to sign Robertson last summer, only for the Scotland captain to turn down the approach, saying at the time: "My heart was still in Liverpool and we wanted to stay."
With that chapter now closed, Simeone has renewed his interest, and the Daily Record reports he has directly instructed the club's hierarchy to make a move.
Juventus and Napoli have also been identified as interested parties. Saudi Pro League clubs are expected to make contact. Celtic have been linked but are not considered realistic contenders.
Robertson made his final Anfield appearance on Saturday against Crystal Palace, scoring the second goal that helped Liverpool to a 3-1 win before Mohamed Salah was forced off injured.