British media reported the deal was worth around 12.5 million pounds ($15.5 million). The 21-year-old's transfer is subject to a work permit and international clearance.
Kinsky's signing adds to Spurs manager Ange Postecoglou's options in goal, with first-choice goalkeeper Guglielmo Vicario spending a lengthy spell on the sidelines after surgery on a broken ankle in November.
Spurs have struggled in the league and are currently in 12th place with 24 points from 20 matches, 21 behind leaders Liverpool.