But the Browns and Steelers could both pass on the FBS leader in sacks in the first round because he's one of the biggest medical risks in the draft. Jones has stenosis, a narrowing of the spinal column.
Diagnosed with this condition at USC as a freshman, Jones transferred to Georgia when doctors wouldn't clear him to play for the Trojans. He went on to record 28.5 sacks in 26 games at Georgia.
Jones got a favorable report from neck and spine specialist Dr. Craig Brigham, who's examining players at the combine, various teams told ESPN's Chris Mortensen. But it's been reported that some teams have taken Jones off their draft board because of his medical condition.
Stenosis has cut short the careers of Sterling Sharpe and Chris Samuels. Former Chargers offensive tackle Marcus McNeill slid into the second round in 2006 because teams were worried about spinal stenosis. McNeill played six seasons, going to two Pro Bowls, before retiring prior to the 2012 season.
Jones said he has no concerns about playing with this condition in the NFL.
"Anybody who steps on that field has a chance of getting hurt," Jones said at the NFL combine. "If you think about it like that, nobody would ever play football. For me, I'm just taking advantage of the opportunity. I love this game. I'm passionate about it."
NFL teams have a couple of months to make their final decision on Jones.