The Basque side are interested in extending the Mexican attacker's stint at the Anoeta after he impressed for the club on loan during the course of the season
The Mexico international, who has struggled to impose himself at the Emirates, joined the Spanish club on a season-long loan last summer with the hopes of securing first-team football.
He played a leading role at Sociedad, netting 12 goals in 36 appearances for the club, and Montanier admitted that the Basque outfit will initiate talks with Arsenal with a view of retaining the 23-year-old.
"We want to keep Vela," Montanier told Infosport, before commenting on the club's transfer policy this summer.
"Eighty per cent of our players are Basque. We want to continue to work with our training centre and also make one or two adjustments to the team."
Real Sociedad finished in 11th place in La Liga with 47 points from 38 matches.