The highly-rated Mexican is not allowed to play for the Gunners due to a work permit problem, but a loan move to Salamanca allowed him to experience European football last season.
The 18-year-old was brought to Europe in 2005 when Arsene Wenger paid £2.5 million to Guadalajara for the youngster's services.
Vela netted eight times for the Segunda Division club last term and he went on to star for Mexico in the Under-20 World Cup in Canada this summer.
Drenthe keen to make Madrid move
Osasuna finished 14th in La Liga last season but enjoyed a run to the semi-finals of the UEFA Cup, where they were knocked out by eventual winners Sevilla.