The Mexican international currently plies his trade with Real Sociedad, and has been a key player for the Spanish side since leaving the Emirates Stadium in 2012.
The 24-year-old played a mayor role in his side’s shock Champions League qualification-winning campaign last season, and has continued to hold a prominent role within their starting XI this term.
His fine displays have alerted Saints chief Mauricio Pochettino, who is keen to bolster his attacking options this winter.
The Argentine has seen his side make an excellent start to the season, and is looking to keep them in the battle for a top four finish by making some top quality additions to his squad.
Vela is seen as the ideal man to slot into their fluid attacking set-up, with his experience of English football also attractive to the south coast outfit.
Although he is a valued member of Real Sociedad’s set-up, the Basque side may be willing to cash in after dropping out of European competition following their dismal fourth place finish in the Champions League group stage.