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