Head coach Mauricio Pochettino has remained patient in the transfer market this summer, while at the same time admitting that he was keen to strengthen his squad before the end of the window.
Earlier on Friday, the Argentine revealed that he wanted to sign four new players, and the first one of those is highly-rated Sanchez, who will pen a six-year contract in North London.
The completion of the transfer is subject to a medical being passed and a work permit being granted, but Sanchez has already spoken of his delight having agreed terms with the Premier League side.
The 21-year-old told the club's official website: "I would like to thank the coach, management, players and fans of Ajax. It is a great move for me to come here to develop my career.
"I am very excited to be signing for such a famous club as Tottenham Hotspur. I am looking forward to working with Mauricio Pochettino and the players and meeting everyone there."
The Colombian international - who had previously been linked with the likes of Manchester United, Chelsea and Barcelona - leaves Ajax having netted six times in 44 appearances in all competition, as well as playing a key role in their run to last season's Europa League final.