Terms of the deal aren't yet known, and it's pending a physical.
Barnette and Matt Bush were the only Rangers relievers with at least 50 appearances in each of the last two seasons.
Minor became a free agent last month when he declined his $10 million option to stay with the Royals.
"During my short-lived free agency, it was hard to imagine an immediate future without being in a Rangers uniform", Barnette said. He was briefly a free agent after not being offered a major league contract before Friday's non-tender deadline.
Gose did not pitch after July 2 because of a strained elbow ligament. He joins starting pitcher Doug Fister as the second addition to Texas' pitching staff this offseason.
Gose hasn't appeared in the major leagues since the 2016 season as a member of the Detroit Tigers when he batted.209 in 30 games. The Phillies originally drafted him as an outfielder out of Bellweather (Calif.) High School in 2008. A scout with an American League club said Gose had a consistent high-90 miles per hour fastball. Gose went 0-2 with a 7.59 ERA in 11 appearances at Class A Advanced Lakeland before being shut down on July 2 with elbow inflammation. After six seasons in Japan, the 34-year-old Barnette signed a two-year deal with the Rangers in December 2015.
The Rangers also signed infielder Christian Lopes out of the Blue Jays' organization. His entire professional career has come with the Toronto organization, having spent the majority of the year at Triple-A Buffalo, where he batted.261 with a.349 on-base percentage, 33 extra-base hits and 18 stolen bases. Given his success last season out of the bullpen, Minor is certainly in line for a multiyear contract. The Rangers will be looking to use him as a starter, like he was previously in Atlanta.