Both clubs had long been in talks over the possibility of the Belgium defender moving to White Hart Lane, and the deal is now set to be completed at last.
Vertonghen will ink a long-term contract with Spurs upon completion of his medical.
"I'm very happy that the saga's finally over. I don't have to worry about anything any more, and can focus on other things again," the 25-year-old commented on his switch.
"It's great that I can start at Spurs during pre-season. That was very important for me. I'm looking forward to a great time. I've heard a lot of beautiful stories about Tottenham."
Vertonghen came through the ranks of the Ajax youth academy, and has made 220 official appearances for the Eredivisie champions, scoring 28 goals in the process.