The club is said to be interested in signing the Mexican striker, whose current deal expires in 2016, until 2018 or 2019.
With the 25-year-old's current contract set to expire June 30, 2016, his club is prepared to offer the prolific forward an improved salary in a bid to fend off interest from rival teams. La Real has reportedly initiated negotiations with Vela's agent, Eduardo Hernandez, in the hope that he will sign an extension in the coming weeks.
Atletico Madrid is rumored to be considering Vela as a replacement for Diego Costa should the Spanish international leave in the summer. Vela's former club Arsenal has the option to activate a 4 million euro buy-back clause in his contract but at present has no plans to do so, Mundo Deportivo reported.
Having joined on loan in 2011 and signed permanently a year later, Vela has proven invaluable at Real Sociedad, which currently sits sixth in La Liga.
The Mexican has appeared in 33 of his team's 34 league games this season, scoring 13 goals and contributing 11 assists in the process. He has compiled 39 goals and 28 assists in his three league campaigns at La Real.