Reds pitching coach Bryan Price watched a Mahay throwing session and recommended him to Jocketty.
The 40-year-old Mahay has a 3.83 ERA over 14 seasons with eight clubs since 1997. He was last in the Majors with the Twins in 2010 and split 2011 in the Minors with the D-backs and Cardinals organizations.
"We'll give him an opportunity to see if he fits with the club," Jocketty said. "If not, he could probably go to Triple-A."
Zavada spent all of last season at Arizona's Double-A club in Mobile, going 3-2 with a 3.43 ERA in 44 games. In 49 big league games over his career, all in 2009, he was 3-3 with a 3.35 ERA.
Cincinnati also named Mark Riggins its Minor League pitching coordinator to replace Mack Jenkins, who earlier this month was promoted to assistant pitching coach on the big league staff. The 54-year-old Riggins spent last season as the Cubs pitching coach.
Mark Sheldon is a reporter for MLB.com. Read his blog, Mark My Word, and follow him on Twitter @m_sheldon.
This story was not subject to the approval of Major League Baseball or its clubs.