Sky Sports News understands the agreement includes a 10 per cent sell-on clause.
The full-back was given permission to travel to Italy on Friday for a medical as negotiations between the two clubs continued. A deal for a total package of £30m has now been reached.
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Spence was a star of England's third-place finish at this summer's World Cup but was likely to find his opportunities limited at Spurs in the coming season, with Pedro Porro the first-choice right-back and Destiny Udogie, Andy Robertson, Souza and Ben Davies all able to play left-back.
Inter have been looking for a new option at right-wing-back and turned their attention to Spence after struggling to strike a deal for Al Ittihad's Moussa Diaby.
Spence had three years left on his contract at Spurs, with an option for a further year, but was keen on the move after enjoying a six-month loan spell with Genoa in Italy from January 2024.
Spurs signed Spence from Middlesbrough in a deal worth up to £20m in July 2022.
The defender took to X to pen a farewell to the club, writing: "A special chapter comes to an end.
"It's been some journey, with plenty of memories, moments and experiences that I'll carry with me for forever, especially that night in Bilbao.
"Thanks to all the staff members who've helped me along the way, the players who've become family and most importantly thank you to the fans.
"It's time for a new chapter, but Spurs will always have a place in my heart. COYS!!"
Spurs already eyeing Spence replacements
Sky Sports News' Mark McAdam:
"Spence is a player Spurs fans will say they have seen his brilliance in flashes. You saw his brilliance at the World Cup with England.
"They may also think it is frustrating. They paid £12.5m for him from Middlesbrough in 2022, a deal that Daniel Levy did. They are getting more than two times what they paid for him.
"But they will also say it doesn't feel like a lot of money for a player who could be so important. Spurs have been preparing for this, though, and already have potential replacements in mind."
