The 28-year-old joined Spurs from Burnley in June 2015 and has played more than 100 times for the club, with his last appearance coming in the Champions League final in June.
He later admitted he was not sure he would see out his contract, which was due to expire in 2022.
Trippier would become Atletico’s third major defensive signing of the summer.
Coach Diego Simeone has already recruited centre-back Felipe from Porto and left-back Renan Lodi from Brazilian side Athletico Paranaense.
With Portugal striker Joao Felix brought in for £113m and midfielder Marcos Llorente arriving from city rivals Real for about £36m, Trippier’s arrival would bring Atletico’s total transfer spend so far this summer to in excess of £170m.
The La Liga runners-up did however receive £107.7m from the sale of France striker Antoine Griezmann to champions Barcelona.