Burton, 28, joined the Bears prior to the 2018 season on a four-year deal that contained $22 million in guarantees.
He played only eight games last season before suffering a season-ending calf injury, and he had hip surgery to address a labrum issue in December. He finished the season with only 14 receptions for 84 yards and no touchdowns.
He is expected to be ready before training camp, a source told ESPN's Adam Schefter.
Burton started all 16 regular-season games during his first year with the Bears, catching 54 passes for 569 yards and six touchdowns, but he was a surprise inactive for the wild-card loss to the Philadelphia Eagles with a groin injury.
He later underwent sports hernia surgery prior to the 2019 season, but he was slow to recover and experienced a setback in training camp.
The Bears open up $6.8 million worth of salary cap space by designating Burton as a post-June 1 cut, but Chicago is still on the hook for the $4 million guaranteed left on his contract.
Burton entered the league as an undrafted free agent with the Eagles. The tight end spent four years primarily as a backup in Philadelphia, but he's remembered for throwing a touchdown pass to quarterback Nick Foles on a trick play -- named the 'Philly Special' -- in the Eagles' Super Bowl LII victory over the Patriots.
The Bears, who had the NFL's least productive group of tight ends in 2019, have eight tight ends under contract following Burton's release.
Chicago added veteran Jimmy Graham in free agency on a two-year deal that included $9 million guaranteed and a no-trade clause. Former Cleveland Browns blocking tight end Demetrius Harris, who spent several years with Bears coach Matt Nagy in Kansas City, also agreed to a one-year deal with Chicago in February.
The Bears are also at the crossroads with 2017 second-round pick Adam Shaheen, a former Division II college player who has yet to emerge as a threat on offense.
In other news, the Bears signed former Green Bay Packers offensive tackle Jason Spriggs to a one-year contract. Selected by the Packers in the second round of the 2016 NFL draft, Spriggs started just nine career games for Green Bay and spent the entire 2019 season on injured reserve with back and elbow issues.