Spurs have been working on the deal since the end of last season, and it appeared that after coming close to deal could have fallen through.
However, the north London club announced on Sunday that they agreed a deal with the Dutch club for the Belgium international, subject to a medical.
Ajax announced in a separate statement that the fee agreed is 12 million euros (£9.5m).
The 25-year-old centre-back enjoyed the best season of his career last season. Named Ajax captain at the start of the campaign, Verthonghen scored seven goals at he led the Amsterdam club to retain their Eredivisie title.
Subsequently he was voted the Dutch league's Player of the Year.
In all, Verthonghen has made 220 appearances and scored 28 goals for Ajax since his debut in 2006.
Should his medical and contract negotiations proceed without a hitch, Vertonghen will become the second major new addition at White Hart Lane this summer, following the signing of midfielder Gylfi Sigurdsson from Hoffenheim.