Trippier was part of the England side which reached the semi-finals of the 2018 World Cup, scoring from a free kick in their 2-1 defeat by Croatia in the last four.
The Spanish giants are paying a fee of €24m to land the England global right-back from last season's Champions League finalists.
Espanyol recently rejected a €20m but Atletico sent an ultimatum this week which saw the Catalans lower their demands and accept a final offer as the player only had one year left on his contract and wanted to move to the Wanda Metropolitano.
The Spanish capital now looks set to become his new home, with Atleti seeking a new right-back having released the experienced Juanfran after his contract expired this summer.
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With the sales of Rose and Trippier, Tottenham will be able to fund the purchase of Ryan Sessegnon, with Fulham asking for 50 million euros for the 19-year-old left-back.
Spurs also had a bid accepted for the defender so says the report, but Saliba has chosen a move to the Gunners over Spurs.
The Spanish club announced the arrival of Trippier on a three-year deal Wednesday, saying he was a "fantastic and experienced" addition to the squad.
"Ben Davies, Eric Dier and Cameron Carter-Vickers will continue their rehabilitation at Hotspur Way", a statement on the official Tottenham website read.
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Trippier may not be the only full-back leaving Tottenham this summer.
Spurs confirmed that Rose has been granted time off "to explore prospective opportunities with other clubs", along with striker Vincent Janssen and winger Georges-Kevin Nkoudou.
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