League sources told ESPN NFL Insider Adam Schefter the pick is a seventh-rounder.
Price, a second-round selection out of UCLA in the 2010 draft, has played in 20 games (14 starts) in his first two seasons, collecting 27 tackles and three sacks.
"You can't have too many good defensive linemen," Bears coach Lovie Smith said of the deal. "As for as what position he'll play, we'll get him into the mix inside -- that's about all I know right now. We'll let (defensive coordinator) Rod (Marinelli) and (defensive line coach) Mike Phair get their hands on him and see if he can help our ballclub."
Price has been dealing with a hamstring problem, so his physical condition is a question mark.
Price was completing a conditioning test for Tampa Bay coach Greg Schiano on Thursday when he heard and felt a painful pop, likely because of scar tissue, the Tampa Bay Times reported. That was enough to prompt the Buccaneers' trade.
"We're kind of thin up there right now," Bears linebacker Brian Urlacher said Thursday. "I'm not sure who our fourth guy is going to be. We lost (Anthony Adams), so we have to find someone to replace him."
Phil Emery, Chicago's first-year GM, echoed Urlacher that Thursday's trade should improve the team's depth at defensive tackle.
"It's a statement that we're going to continue to work on our roster," Emery said. "We're never going to rest on our roster, to find players that we feel can help us in our scheme and push towards our goals."
Price was at the center of a fight with rookie Mark Barron, the Bucanneers' first-round draft pick, during the team's offseason program, the Tampa Bay Times reported.
Schiano confirmed the fight to the newspaper last week but didn't expect any lingering issues.
"I hope not," Schiano told the Times. "We all make mistakes. Be men about it and let's go. Ultimately, everything we do has one goal and that's to bring that trophy back here. We have to put our own personal preferences and whatever disagreements we have aside to do what's best for the club."
Price's agent also told the Times in May that Price missed organized team activities after being hospitalized for exhaustion and grief in the wake of his sister's death.
Price's 30-year-old sister, Bridget, was killed in an automobile accident.
"He has struggled some here in the last year," Emery said. "He's had a tough go of it with family issues, obviously with his sister passing away. He's had a hard time with the transition. He's a player that needs a new home and we're excited to bring him to Chicago. "
With the move, the Bears' roster now stands at 90 players.
Information from ESPNChicago.com's Michael C. Wright was used in this report.