Jones-Drew will not need surgery, which means he does not have a Lisfranc injury, the source said.
On Monday, the team ruled Jones-Drew out for its Week 8 game against the Green Bay Packers.
Jones-Drew suffered the injury to his left foot on Jacksonville's first play from scrimmage in Sunday's 26-23 overtime loss to the Oakland Raiders. The seven-year veteran was standing on crutches on the sideline in the second half and left the stadium in a walking boot.
The Jags also are dealing with an injury to starting quarterback Blaine Gabbert, who left Sunday's game with an injury to his non-throwing shoulder.
A source told ESPN NFL Insider Ed Werder that an MRI exam on the second-year quarterback showed he has a torn left labrum.
Jones-Drew, who leads the Jaguars with 414 rushing yards, was replaced by Rashad Jennings, who ran for 44 yards against Oakland and will start against Green Bay.
Toston spent the preseason with the Jaguars before being waived in final cuts. He ran 22 times for 171 yards, including a 42-yard touchdown at New Orleans. Toston spent the 2010 season and the 2011 preseason with St. Louis and will back up Jennings on Sunday.
Toston also was a four-year letterman at Oklahoma State, where he finished with 2,757 yards rushing and 27 touchdowns.
Information from The Associated Press was used in this report.