Mexico international Vela spent last term on loan at Sociedad and has previously made it clear that he would prefer not to return to the Emirates.
The 23-year-old has never been able to establish himself as a regular for the Gunners and would not appear to be at the forefront of Arsene Wenger's thinking for the upcoming campaign.
A departure from north London could therefore be on the cards, and Sociedad are now hopeful that something could be resolved in the next few days.
"The president is optimistic because I want a quick solution," said Montanier. "I expect that Carlos will arrive in this next week."
Reports suggest the latest deal would be another loan with an option to buy, and Sociedad president Jokin Aperribay says discussions are progressing well.
"We're getting somewhere, but there's nothing set in stone yet," Aperribay told AS.
"As the talks have evolved, I'm moderately optimistic, but there's no deal yet."