Headley was injured Sunday while sliding into second base trying to break up a double play. He went to San Diego, and an exam Monday revealed the fracture.
The injury is not likely to require surgery and will need about a month to heal.
"Injuries occur. You hate to see it happen, but it's the nature of a physical sport like baseball," Padres manager Bud Black said. "It's unfortunate for Chase and the entire Padres organization."
Headley hit .286 with 31 homers and 115 RBIs last year, when he won a Gold Glove and finished fifth in NL MVP voting.
Logan Forsythe could take over at third. Second baseman Jedd Gyorko, the Padres' top prospect, played third base until switching positions last season in the minor leagues.
Prospect James Darnell is out at least two weeks with an oblique strain.
"We're going to sort that out," Black said. "This time of the year is important as we finalize our roster. We'll work through today's bad news. We feel as though we have some depth on the infield, and we'll react accordingly."