The forward is understood to be returning to South Korea for surgery following a heavy tackle by Aston Villa's Ezri Konsa in the opening minutes of Sunday's 3-2 win.
The injury is thought to be significant, with Son set to be sidelined for months, but Spurs will not know his full prognosis until after the surgery.
In a statement, the club said: "Following surgery, our medical staff shall be reviewing management options for Son’s rehabilitation with the player expected to be sidelined for a number of weeks."
Son received treatment on the pitch at Villa Park but played on and scored twice, including a stoppage-time winner when he showed no ill-effects as he raced away from the Villa defence. He was later holding his arm during a number of post-match interviews and a scan on Monday revealed the extent of the injury.
The loss of the South Korean is another major blow to Jose Mourinho and Spurs, who are already without Harry Kane and Moussa Sissoko until April.
Mourinho, the Spurs manager, has even raised the possibility that Kane, who had surgery on ruptured hamstring tendons in January, could miss the remainder of the season.
In the absence of the England captain, Son has stepped up, scoring six times in his last five appearances, including the double at Villa Park. In total, the South Korea international has scored 16 goals this season and he became the first Asian player to reach 50 Premier League goals on Sunday.
Spurs failed to sign a centre-forward to cover for Kane in January and have been left with Son and Lucas Moura to deputise up-front, despite Mourinho insisting they are "not strikers".
The club did recruit winger Steven Bergwijn from PSV Eindhoven last month and the 22-year-old Dutchman will now have to step up, along with Erik Lamela, Ryan Sessegnon and teenage forward Troy Parrott, 18.
After the win at Villa Park, Mourinho warned that Son, who was one of the only Spurs players not to suffer an injury last season, and Lucas were in danger of burnout given their work-load but the Portuguese's options as now even further diminished, as he targets a place in the Premier League top four and another Champions League run. Spurs face German club RB Leipzig in the last-16 first leg on Wednesday night, with Mourinho due to address the media at Hotspur Way this afternoon.