Kinsler will be a free agent after the 2018 season, and the Angels are expected to assume all of the remaining $11 million on his contract.
Last season, he batted a career-worst .236 with 22 homers and 52 RBIs in 139 games. Despite the low batting average, Kinsler was among five second basemen to hit 20 homers and 25 doubles and score 90 runs.
Kinsler's contract contained a no-trade clause. Detroit general manager Al Avila told reporters that among teams interested inquiring him, Kinsler was willing to waive it to join the Angels.
Kinsler has appeared in at least 135 games for seven straight seasons and is a four-time All-Star. He also won a Gold Glove in 2006.
Since making his major league debut in 2006, Kinsler owns a career .273 average with 234 homers and 839 RBIs for Texas and Detroit. He also has a .291 average in 37 postseason games.
The Angels, who missed the playoffs for the third straight season, sent outfielder Troy Montgomery and right-hander Wilkel Hernandez to Detroit.
Montgomery, 23, batted a combined .271 with a .358 on-base percentage in low Class A, high Class A and Double-A last season. He was the Angels' eighth-round pick in 2016.
Hernandez, 18, went 4-1 with a 2.64 ERA for two rookie-level teams last season. The Angels signed him as an international free agent in 2015.