The Belgium international had two years left on his contract but was keen to leave this summer and has moved to the Middle East for a fee believed to be £13million.
Alderweireld made 236 appearances for Spurs after joining from Atletico Madrid in 2015 and was an integral part of the side that mounted back-to-back Premier League title challenges and got to the Champions League final.
"Thank you first to the fans and then to my team-mates and every incredible member of the Spurs staff, backroom and beyond. This club will always be in mine and my family's heart."
Spurs are looking to replace Alderweireld and are in negotiations with Atalanta for Christian Romero.
They have already brought in Atalanta goalkeeper Pierluigi Gollini and Sevilla winger Bryan Gil, with Lamela moving the other way.