Following the appointment of Nuno Espirito Santo, Spurs are stepping up their efforts in the transfer market.
They are close to signing Atalanta goalkeeper Pierluigi Gollini on loan and new director of football Fabio Paratici wants to bring in Gil too.
Talks are underway with Sevilla over a deal for the 20-year-old winger, who is part of the Spain squad competing at the Tokyo Olympics.
A deal for Gil is likely to involve a fee of just over £20m, plus Lamela moving to Sevilla too.
Lamela, who has been with Tottenham since 2013, is thought to be ready for a new challenge and the promise of regular first-team football.
Gil spent last season on loan at Eibar and impressed during his first full season playing in La Liga.