The reported deal is pending a physical and the team plans to play him at multiple positions, according to Heyman - including shortstop.
Gonzalez, 29, will receive $12 million this year and $9 million in 2020, a source told The Associated Press.
Minnesota also made a managerial change after the club failed to meet expectations last season, replacing Paul Molitor with Rocco Baldelli.
The Twins finished 78-84 in 2018 after reaching the American League wild-card game in 2017. Their revamped lineup could potentially challenge the Cleveland Indians for the AL Central crown.
Following the reported agreement, Gonzalez's former Houston Astros teammates shared their thoughts on Twitter:
Gonzalez hit 16 home runs and slashed a modest .247/.324/.409 in 145 games last season for the Astros. However, the Venezuelan hit 23 homers with 90 RBIs and a .907 OPS in 2017 while helping Houston win the World Series.