According to BBC Sport, the 32-year-old left-back will officially join Spurs on July 1 after the expiration of his contract at Liverpool, ending a successful seven-year spell at Anfield.
Robertson had earlier announced his decision to leave Liverpool this summer, paving the way for Tottenham to secure his services without paying a transfer fee.
The North London club had attempted to sign the defender during the January transfer window. Although an agreement was reportedly reached between both clubs, the move collapsed after Liverpool failed to recall Kostas Tsimikas from his loan spell at Roma.
Tottenham manager Roberto De Zerbi welcomed the arrival of the experienced full-back, describing him as a player with proven quality and leadership.
“Andy is someone I’ve admired for a number of years and he will bring outstanding technical qualities, experience, leadership and mentality to our team,” De Zerbi said.
“He is a proven winner at the highest level over a long period and is someone who can be a big player for us, both on and off the pitch.”
Robertson joined Liverpool from Hull City in 2017 and made 378 appearances for the Merseyside club.
During his time at Anfield, he won two Premier League titles, the UEFA Champions League, two League Cups and the FA Cup.
The Scotland international remains a key figure for his national team and is expected to captain the side at this summer’s World Cup. He has earned 92 international caps.
Tottenham sporting director Johan Lange also praised the signing, highlighting Robertson’s winning mentality and experience.
“His quality, character and leadership have been evident throughout a career in which he has regularly competed for – and won – major honours,” Lange said.
“Andy’s professionalism and commitment will also be invaluable to the development of our squad, and he shares our ambition and determination to bring success back to the club.”
Robertson is among Tottenham’s first major additions ahead of the new season as De Zerbi seeks to strengthen the squad and improve the club’s fortunes in the Premier League.
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