CBS Sports' Jason La Canfora first reported the news.
The Cardinals also released defensive lineman Ryan McBean.
Kelly, 33, was cut by the New England Patriots on Sunday. He had 2.5 sacks in five games last season, his first with New England, before he tore his ACL last October.
The 6-foot-6, 300-pound lineman spent the first nine seasons of his career with the Oakland Raiders. In 132 career games, he has 37.0 sacks and 301 tackles.
Arizona linebacker John Abraham missed the first three weeks of training camp following an offseason DUI charge, and safety Tyrann Mathieu is returning from a torn ACL.
The Cardinals finish their preseason schedule on Thursday against the San Diego Chargers. They open the regular season at home on Sept. 8 against the Chargers.
- Paul Palladino