Gil spent the second half of last term on loan at Valencia after failing to win over Antonio Conte and again this summer his future is under the scanner.
He has been linked with another loan exit, although his recent displays in training with Tottenham have left the door open to a stay.
Valencia are keen to take him back and, according to Italian broadcaster Sportitalia, Gattuso sees Gil as his main target.
The Italian tactician wants to have the winger added to his squad before the transfer window slams shut.
And Valencia are hopeful that they can reach an agreement to re-sign Gil, who wants the move.
The 21-year-old winger was an unused substitute in both Tottenham’s two Premier League games this season.
Spurs boss Conte has been clear that if any players in his first team squad depart then he would expect them to be replaced.
