The Mexico international has found himself out of favour at Emirates Stadium in recent years, forcing him to take in regular loan spells.
The 23-year-old striker has spent time with Salamanca, Osasuna, West Brom and Real Sociedad.
A successful season in Spain last term saw him produce his best return in front of goal.
Sociedad have now expressed a desire to bring him on board full-time, and Arsenal have granted permission for transfer talks to be held.
Vela has been on the Gunners' books since 2005, but has been restricted to just 62 senior appearances during his time in North London.
"Arsenal Football Club has announced that striker Carlos Vela is in San Sebastian this evening for final discussions about a permanent move to Spanish La Liga side Real Sociedad," read a statement on Arsenal's official website.
"The 23-year old Mexican international spent the whole of 2011/12 on loan with the Spanish club and enjoyed a successful campaign in La Liga, scoring 12 times in 35 appearances.
"Vela made his Arsenal debut in 2008 and has made 62 appearances, scoring 11 goals."
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