Jones failed the team's conditioning run -- which entailed running two sets of 10 sprints of 50 yards in less than seven seconds -- as did safety Brodney Pool.
"I don't want to get into the reasons why any of those guys were not able to do it, the fact is, they didn't do it," Garrett said.
Jones is the backup to DeMarco Murray, who is returning from an ankle injury but led the team with 897 yards on 164 carries last season, including games with 253 yards, 139 yards and 135 yards.
Jones was inactive for four games in the middle of the season, beginning with the Oct. 23 game against the Rams in which Murray exploded for a franchise-record 253 yards on 25 carries and a 91-yard touchdown.
Drafted in the first round (22nd overall) of the 2008 draft, Jones is entering the final year of his contract. The 25-year-old has never started more than eight games in a season but averaged 5.1 yards per carry in his first four years in the NFL.