Austin fouled a ball off his left foot early this week taking batting practice during a pre-spring training workout at the minor league complex, and manager Joe Girardi said an MRI Friday found a small bone break.
Austin will be in a boot for three weeks and won't start baseball activities for six weeks.
"It's unfortunate," Girardi said.
Carter, who tied for the National League in home runs last season, with 41, finalized a $3.5 million, one-year contract Thursday. A first baseman, the 30-year-old Carter is likely to see time at designated hitter.
OF Jacoby Ellsbury will not join the team until early next week due to the pending birth of his child. Position players are scheduled to report Saturday. OF Mason Williams has left patella inflammation and will be shut down for two weeks.