"We're extremely excited about the progress he has made," Bradley told reporters.
The 28-year-old Jones-Drew missed 10 games in 2012 because of a Lisfranc injury to his left foot. The running back underwent surgery in December and spent several months in a walking boot.
"Now, it's back to football," Jones-Drew told reporters. "It's been a long time coming. I'm excited to be out there running around and playing."
He said he was training during his rehab "like I wasn't hurt."
Bradley also said defensive end Jason Babin, who had groin surgery last month, has been cleared for camp. Four players, however, will begin training camp on the physically unable to perform list, including receiver Justin Blackmon, who had groin surgery earlier this month, and rookie safety Johnathan Cyprien, who strained his hamstring last week.
Bradley said the Jaguars hope Cyprien will be able to practice with the team next week but Blackmon's return "might be a little longer."
Jones-Drew was admittedly out of shape when the team's offseason program began in April, but he appeared to be in considerably better shape when he returned for a three-day minicamp with the Jags last month.
Jones-Drew said he weighed approximately 230 pounds at the beginning of the team's offseason program and hopes to get down to 212 to 215 pounds by the start of the regular season.
"I don't need to drop much, but just a little bit more just so I get comfortable and get ready to play," he said.
Despite being away from the game for the last 10 months, Jones-Drew said he hasn't changed since he last carried the ball for the Jaguars.
"I'm the same guy. That's how I've been training, like I wasn't hurt," he said. "I just can't wait to get back out there. "For me, not playing football for 10 months, not wearing pads, not having that soreness, that's the thing you want to get back and get used to. Other than that, everything else is natural."
Jones-Drew also said the offseason incident at a St. Augustine restaurant is behind him. He allegedly hit a bouncer in the face at The Conch House Restaurant in late May. Jones-Drew was never officially charged.
"It has been behind me. It was just an unfortunate situation that happened," he said. "I'm just ready for football now. That was a hot story for a while, now it's time to play football. It's a relief now that we can talk football."
The 5-foot-7 dynamo has been the centerpiece of one of the league's worst offenses the past four years, running for 4,735 yards and 29 touchdowns against stacked fronts.
In seven years with the Jaguars, he has 7,268 yards rushing, 2,559 yards receiving and 76 total touchdowns.
He enters this season on the final year of a five-year, $31 million contract but said he isn't thinking about his looming free-agent status next offseason.
"Every year is a big year. You take care of what matters first, which is on the field, and everything else has its place. That's all I'm going to do," he told reporters.
Information from The Associated Press was used in this report.