The 20-year-old Gil is currently representing Spain at the Olympics.
Gil, 20, has agreed a contract until 2026 at the Tottenham Hotspur Stadium.
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 在八月 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.”