Cundiff, who worked out Monday for team officials, will compete with incumbent Nick Folk. Training camp opens Thursday in Cortland, N.Y.
Statistically, Folk enjoyed his best year with the Jets in 2012, converting 77.8 percent of his field goal attempts, but the coaching staff has always felt he performs best when pushed by a veteran. A year ago, he beat out Josh Brown in training camp.
Cundiff, 33, who has kicked for five different teams, enjoyed his best years with the Baltimore Ravens from 2009 to 2011. But his time with them ended horribly, as he missed a 32-yard field goal in the final seconds of the 2011 AFC Championship Game.
Cundiff spent five games last season with the Washington Redskins, but finished the year with the San Francisco 49ers. He was signed as insurance during their playoff run when David Akers slipped into a funk, but Cundiff never got into a game.
To make room for Cundiff, the Jets waived kicker Brett Maher, a first-year free agent.