English newspaper the Daily Telegraph said Trippier, 28, has travelled to the Spanish capital ahead of a move worth 20 million pounds ($24.83 million).
Spanish daily Marca said Atletico, who finished runners-up in La Liga last season, have turned to Trippier after failing to sign Portugal right back Nelson Semedo from Barcelona.
An source from Atletico told Reuters that negotiations were ongoing with Tottenham.
Trippier helped Spurs to the Champions League final for the first time in their history last season, where they were beaten 2-0 by Liverpool in Atletico's Wanda Metropolitano stadium.
Speaking after the final, the right back admitted his future with Tottenham was unclear, saying: "I want to stay in England, but what can you do? I'll see what happens and what the club wants, most importantly."
Trippier was part of the England side that reached the last four of the 2018 World Cup, scoring from a free kick in their 2-1 semi-final defeat against Croatia.
Marca also said that Atletico are set to sign Spain defender Mario Hermoso from Espanyol as they look to bolster their back line after a spate of high-profile departures.
Atletico bid farewell to former captain Diego Godin and fellow stalwart defenders Filipe Luis and Juanfran after their contracts expired.
In January they agreed to sell France defender Lucas Hernandez to Bayern Munich for 80 million euros ($89.7 million).
Striker Antoine Griezmann, Atletico's top scorer in each of the past five seasons, has left for champions Barcelona for 120 million euros, with holding midfielder Rodri also departed after Manchester City paid his 70 million euro release clause.
Atletico have already brought in 30-year-old central defender Felipe from Porto and 21-year-old left back Renan Lodi from Athletico Paranaense.
They have also increased their attacking options by landing Portugal forward Joao Felix from Benfica for a club record 126 million euros as well as signing Serbian forward Ivan Saponjic, Mexico midfielder Hector Herrera and Spanish midfielder Marcos Llorente.