Barth will sign a one-year deal with the Bears on Monday, a source told ESPN's Adam Schefter.
The 30-year-old Barth spent training camp and the preseason in New Orleans but lost the kicking competition to Kai Forbath. Barth was cut by the Saints over the weekend.
In 12 games for the Tampa Bay Buccaneers in 2015, Barth made 23 of 28 field-goal attempts (82.1 percent). For his career, he has an 84.8 percent accuracy rate on field goals.
Barth kicked five games for the Broncos in 2014 when Bears coach John Fox was in Denver.
The 34-year-old Gould exits Chicago with a franchise-best 1,207 points. He has spent his entire 11-year NFL career with the Bears, making 276 of 323 field-goal attempts. His 85.4 percent success rate ranks ninth in league history. Last year, however, he was just 21-for-27 (77.8 percent), and he missed two extra points this summer in preseason action. That was enough to convince the Bears not to pay his $3 million salary.
Gould thanked the fans and organization on Twitter late Sunday night.
"Thank you to the Bears organization, teammates, coaches, and fans. Once a Bear, always a Bear. #beardown," Gould tweeted.
The Bears also made a change at long snapper, re-signing Patrick Scales and releasing Aaron Brewer.