The England international has chosen to join Spurs ahead of a rival offer from Aston Villa.
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The Athletic says only minor details need to be ironed out with Spurs organising a medical for Gallagher some time this week.
Gallagher joined Atletico from Chelsea for &eurio;38m in August 2024.
Spurs are expected to splash out around €40m to bring the midfielder back to London this week.