The 21-year-old is being monitored by several teams this summer.
According to Fabrizio Romano, Sampdoria are the keenest to secure his signature, but Marseille are also involved in talks.
Gil excelled at Valencia during the second half of last season on loan.
There has been no decision about his short-term future, per the source, with Spurs considering their options.
Gil is not thought to be in the immediate plans of coach Antonio Conte for the coming season.