Gil, 20, has agreed a contract until 2026 at the Tottenham Hotspur Stadium.
He will link up with his new team-mates at the conclusion of the Olympics in Tokyo, where he is representing Spain
Tottenham are reported to have paid around £21million for Gil, as well as letting Lamela head in the opposite direction.
The move brings an end to Lamela’s eight-year stint at Spurs
The 29-year-old Argentinian joined from Roma in August 2013 and made 257 appearances in all competitions.
Spurs said on their website: “A club man through and through, we wish Erik all the best for the future.”