The 19-year-old, who only stepped up to the club's senior team in January this year, is one of the most highly-rated young defenders in world football.
Spurs are believed to have been in talks with Estudiantes for some time, but faced competition from Paris Saint-Germain, while Inter Milan and Roma were also reportedly interested.
According to Sky Sports News, the North Londoners have won the race to sign Foyth, quoting an interview given by the player to an Argentine newspaper in which he says compatriot Mauricio Pochettino was instrumental in convincing the youngster.
Foyth is quoted as telling El Dia: "It's a club that bets on the young and I think I'm going to have a chance to play.
"I also spoke with Mauricio Pochettino and I liked what he said. It was what helped me decide that Tottenham was the right place to continue my career.
"He told me to go, train and that I would be involved I would play. He also told me what the club was like, what job idea it had."