The Mexican attacker has brought to an end his seven-year stint with the Premier League club in order to sign a permanent deal with the Spanish outfit
The Mexican striker was on loan with the Spanish outfit last season, finishing as La Real's top scorer with 12 goals and making seven assists in 35 appearances.
Vela has previously played on loan at Salamanca, Osasuna and West Brom after failing to nail down a place in the Arsenal first team since joining the club in 2005.
The 23-year-old is expected to put the finishing touches to the deal on Tuesday, though Sociedad have already announced an agreement, with an official presentation set to take place on Thursday.
Vela has represented Mexico at every youth level, playing at the Under-17 and Under-20 World Cups, and was a part of the senior side for the 2010 World Cup in South Africa.