The Spanish giants are paying a fee of €24m to land the England worldwide right-back from last season's Champions League finalists.
But the former Manchester City trainee still made 38 appearances across all competitions as Spurs finished fourth and made it all the way to the Champions League final at Atletico's Wanda Metropolitano Stadium.
Atletico did not immediately respond to a request for comment on the reports from Reuters.
Trippier arrived at Spurs back in 2015 for a fee of around £3.5 million.
A key man in England's run to the last four of the 2018 World Cup, scoring the opening goal in the semi-final defeat to Croatia, Trippier's form dipped during the course of the 2018/19 campaign.
Trippier's departure leaves Spurs boss Mauricio Pochettino to choose from Serge Aurier, Kyle Walker-Peters and Juan Foyth on the right side of defence.
The 18-year-old was initially tipped to join Arsenal in a £27m deal but the Metro claim Tottenham tried to hijack their rivals move by submitting an improved bid and offering Saliba better contract terms.
"Because of all this, we welcome Trippier to the Atleti Family and wish him the best of luck at his new home".
Atletico seem to have won the race for Trippier, and Tottenham are set to make a huge profit on his sale to the Spanish giants.
Pochettino will now have Serge Aurier and Kyle Walker-Peters as right-back options for the season ahead, although the latter has suitors at rival Premier League clubs.