Jaso, 30, met with a concussion specialist in Pittsburgh on Wednesday and was advised to stop baseball activities.
He has been out of the lineup since an Aug. 23 collision with the White Sox' Carlos Sanchez resulted in his second concussion in 13 months.
"Just wasn't happy in the direction that he was going," Melvin said of the assessment by Dr. Micky Collins.
Jaso was hitting .264 with nine home runs and 40 RBI in 99 games before the injury.
Oakland was nine games behind the Los Angeles Angels in the American League West entering Thursday. The Athletics held a 1.5-game lead for the first Wild Card spot.
- Paul Palladino