The D-backs sent the reliever to the Rays as part of a three-team trade that netted Arizona a pair of Minor Leaguers and about $5.5 million in salary relief for 2014.
As part of the deal, the Rays received Bell and $500,000 from the D-backs, and Tampa Bay also got catcher Ryan Hanigan from the Reds.
The D-backs received Minor League pitcher Justin Choate and a player to be named or cash.
The Reds received left-hander David Holmberg from the D-backs.
Bell was acquired by the D-backs last offseason from the Marlins in exchange for infield prospect Yordy Cabrera. The Marlins also included cash in that deal to help offset the $18 million that was left on Bell's contract.
Bell is set to make $9 million in 2014, and he has an option for 2015 that becomes guaranteed if he finishes 55 games next season. The D-backs will pay $500,000 of the contract, with the Marlins still on the hook for $3.5 million and the Rays to pay the remaining $5 million.
The D-backs felt that they could trade Bell given the presence of young relievers Matt Stites and Jake Barrett, both of whom will compete for bullpen spots in Spring Training. In addition, the team is considering moving Randall Delgado to the bullpen if a starting pitcher can be added.
The move also saves the D-backs $5.5 million that they can use toward adding a starting pitcher, corner outfielder and bench help this offseason.
"It helps a great deal," general manager Kevin Towers said of the salary relief. "We've still got some bench pieces that we need to put together. We still have interest in bringing back Eric Chavez, a guy who was a key bench player on our club last year. We're also looking to add depth to our starting pitching, preferably a No. 1, No. 2. More than likely that would probably come via trade, although there are some free agents we have some interest in."
The D-backs probably had less than $10 million to spend this offseason before getting the additional $5.5 million savings. Towers declined to say what the payroll will end up being, but it likely will cross the $100 million threshold for the first time since 2002.
Prospect acquired by D-backs
Justin Choate, RHP: It's not often that a player signs for $1,000 as a non-drafted free agent in June and is traded for a big leaguer six months later, but that's what happened to Choate. After spending two years at Texarkana (Texas) Junior College and two at Stephen F. Austin State, Choate pitched one game for the independent Roswell Invaders (Pecos League) before signing with the Rays. He went 1-3 with a 2.88 ERA and six saves in 16 games at short-season Hudson Valley, posting a 35-9 K-BB ratio in 40 2/3 innings. At six feet and 170 pounds and armed with an 88- to 92-mph fastball, Choate isn't imposing. His best attribute is that he throws strikes, and he's best suited for a relief role. He also throws a curveball and a changeup, but neither grades as an average offering.
Choate made 16 relief appearances for the Rays' affiliate in the New York-Penn League last season. He was signed by the Rays as a non-drafted free agent last summer for $1,000.
"I would say the player to be named later is someone we value as a prospect," Towers said. "Of course, we can't announce who that is yet. That's not taking anything away from Choate. But I'd say probably the key player in the deal, with the players from Tampa Bay, will be the player to be named later. Choate was a guy who [has] only pitched a year, very good strikeout-to-walk ratios, pitched well at the low levels, not an overpowering guy. But he is a strike-thrower, which is good."
Holmberg, 22, was selected by the White Sox in the second round of the 2009 First-Year Player Draft, and he was dealt to the D-backs in July 2010 -- along with right-hander Daniel Hudson -- in exchange for Edwin Jackson.
Holmberg made one start in 2013, allowing three runs in 3 2/3 innings. He was ranked as the No. 5 prospect in the D-backs system by MLB.com. In 26 starts for Double-A Mobile last year, he went 5-8 with a 2.75 ERA.
Jeff Pico, who was the D-backs' field coordinator last season, is now the Reds' pitching coach, and his familiarity with Holmberg likely played a role in the deal.
"We liked Holmberg, as did Cincinnati," Towers said. "He's probably one of those guys who's a power pitcher, but he's always put up good numbers."
Steve Gilbert is a reporter for MLB.com. Read his blog, Inside the D-backs, and follow him on Twitter @SteveGilbertMLB. MLB.com reporters Bill Chastain and Mark Sheldon contributed to this report. This story was not subject to the approval of Major League Baseball or its clubs.