Gabbanini was hired by former director of football Fabio Paratici and took a lead role in their summer recruitment.
The 43-year-old Italian had worked at Watford and Udinese, following his role as head of academy coaching at Fiorentina.
Paratici departed Spurs earlier this year, but Gabbanini stayed put, continuing his work under new boss Ange Postecoglou.
The former chief scout had helped to draft in summer signings James Maddison, Guglielmo Vicario, Ashley Phillips, Micky van der Ven, Manor Solomon, Alejo Veliz, and Brennan Johnson.
Tottenham chairman Daniel Levy has undergone changes since Paratici's departure. Scott Munn has recently been brought in as the club's new chief football officer, with the director of football and chief scout roles in need of being fulfilled.
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Spurs are also set to take up the option to extend Son's contract until 2026 to avoid him running his deal down, talkSPORT understands.
The 31-year-old is into his ninth campaign with the north London club, and has just over 18 months left on his current deal which runs until 2025.
Son was recently named the new captain replacing Hugo Lloris, and remains a key player at the Tottenham Hotspur Stadium.
The South Korea international netted a hat-trick in Spurs' 5-2 win at Burnley earlier in September.
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"Everyone knows he's a fantastic footballer but he's just a brilliant human being," Postecoglou recently said.
"For any sort of manager, when you're looking for leaders in a group and they're all different types, firstly what kind of person are they is really important when setting up a team.
"Even from afar, even before I joined the club, obviously I've known Sonny for a long time because being from that part of the world myself and working in Asia and the exposure I had to his standing in the community there, you can tell he was respected.
"Whether it was teammates or opponents, everyone spoke really highly of him. I think he was the perfect embodiment."