Sevilla, Deportivo La Coruna and Celta Vigo are also said to be keen to bring the Spanish midfielder back to La Liga this summer.
Gil, 23, originally joined Villa from boyhood club Valencia in January 2015 for around £3.2 million and has since made 34 total appearances for the Midlands outfit, recording three goals and three assists.
The diminutive playmaker has shown some tantalising flashes of ability during his 18 months in England, but he also struggled to produce a consistent run of form and failed to hold down a regular place in the side under either Tim Sherwood or Remi Garde this past season as the club finished bottom of the Premier League with just 17 points.
While he remains a popular figure with fans, the former Under-21 international is now looking to leave Villa Park this summer and his agent, Federico Marco, revealed to El Desmarque earlier this week that Celta Vigo had seen an offer for him turned down.
“His intention is not to play in the Championship and look for teams in the Premier League and La Liga,” Marco said of Gil’s plans. “We have not spoken with Celta, but in the past we have been negotiating and we know he is a player to their liking.
"They made an offer for him, but Aston Villa did not accept it."
Nevertheless, despite Villa’s reluctance to sell Gil, whose contract runs until 2019, more Spanish clubs have emerged as possible suitors for his signature since his agent’s comments.
Plaza Deportiva has claimed that Deportivo La Coruna, Sevilla and Betis are also keen to lure him back to La Liga, with their report adding that Villa could be prepared to let him spend the 2016-17 season out on loan as a compromise.
Betis’ new sporting director, Miguel Torrecilla, is believed to have tried to sign the Valencia native last summer while working at Celta, and Plaza Deportiva’s report claims that the Andalusian side would be his likeliest destination in the event of a loan move.
Interestingly, the club’s recently appointed manager Gus Poyet played with incoming Villa boss Roberto Di Matteo at Chelsea between 1997 and 2001, but whether their relationship could help broker a deal between the two sides remains to be seen.