The deal is worth up to $24 million, sources said, and the Cavs hold a team option for the second year. Only $6 million of the contract is guaranteed, according to sources.
Bynum, who missed the entire 2012-13 season with the Philadelphia 76ers because of a knee injury, will have to reach a variety of health- and games-related benchmarks to collect the full $12 million in the first year of the deal, sources said.
Sources said the Cavaliers, however, see the $6 million guarantee as a low-risk move given how rarely a former All-Star center such as Bynum becomes available on the open market -- and how rarely they are available to lottery teams such as Cleveland.
If Bynum exceeds all expectations and plays well, the Cavs also can pick up the second-year option and have another attractive piece alongside star guard Kyrie Irving for their long-planned attempt to bring LeBron James back to Cleveland in 2014 free agency.
Cavaliers owner Dan Gilbert was a force in the pursuit of Bynum and was willing to gamble $6 million that the 25-year-old will have a bounce-back season in 2013-14, league sources told ESPN.
Neither Atlanta nor Dallas, according to sources close to the talks, made Bynum a formal offer after their respective sitdowns with him.
In addition, after a breakfast meeting with Bynum and medical evaluations in Dallas, Mavericks officials came away with what one source described as "significant concerns" about Bynum's ability to be ready for the start of the season or to have any significant impact in 2013-14. That made the Mavericks reluctant in the extreme to try to outbid the Cavs, who had more financial flexibility to make this sort of gamble, sources said.
Although the Cavs are realistic about the severity of Bynum's injury history, the team was encouraged by recent MRI results, league sources told ESPN. The Cavs hope Bynum will be ready for training camp.
Bynum will be reunited with Cavaliers coach Mike Brown, who coached him to his best individual season with the Los Angeles Lakers in 2011-12, when Bynum averaged career highs of 18.7 points and 11.8 rebounds in earning his first All-Star berth.
ESPN The Magazine's Chris Broussard and ESPN.com's Marc Stein and Brian Windhorst contributed to this report.