Cora currently works as a baseball analyst for ESPN and has no professional coaching experience, but he played in the major leagues from 1998 to 2011 for six teams.
The 39-year-old Cora also has been mentioned in connection with manager openings in Washington and Miami.
According to the CBS Sports report, Cora is seen a good managerial candidate despite his lack of experience. A bonus is his ability to speak Spanish.
The Padres' vacancy was created when the team dismissed interim manager Pat Murphy after finishing 74-88 in 2015. Murphy stepped in to replace ousted manager Bud Black in June.
Gardenhire also could be on the radar of the Nationals, who fired Matt Williams at the end of a disappointing 2015 season.