The Colombia centre-back signed a contract until 2023 on Wednesday and is expected to cost Spurs up to £42m including add-ons, subject to receipt of his work permit later this week.
Sanchez told Spurs' official website: "This is a massive opportunity for me because of what Tottenham represent, not just here in England but around the world.
"It's a huge step forward in my career and will allow me to continue developing and achieving big things. This is a great chance."
The 21-year-old played 47 times for Ajax in 2016-17 as the club achieved second place in the Eredivisie and reached the final of the Europa League.