The winger joined Tottenham from Spanish giants Sevilla last summer, but struggled to earn game time in England.
Spurs boss Antonio Conte allowed Gil to return to his homeland in the latter half of last season where he managed to impress with Valencia on loan.
Conte does not see Gil playing a key role under him this season and has given the greenlight for him to find a new temporary home.
The Spaniard is also keen on leaving Spurs to earn game time on a regular basis and he is once again in the sights of Valencia, who see him as a potential replacement for Guedes, a player strongly linked with exiting the club.
And according to Spanish outlet Plaza Deportiva, new Valencia coach Gattuso is open to the idea of bringing Gil back to the club this summer.
Valencia made an approach for Gil earlier this summer but Tottenham refused to loan him then as they wanted to see if they can sell him; the Spanish are tipped to move for the winger again, provided Guedes leaves the Mestalla Stadium.
In addition to Valencia, Gil has also been credited with interest from Serie A side Sampdoria and Real Sociedad.