The Twins hope Gonzalez makes a decision soon, and the club believes it has strong competition to sign the 29-year-old.
The Houston Astros - the team Gonzalez won a World Series with in 2017 - have been linked to the Venezuelan during the winter, along with the San Diego Padres and Washington Nationals.
Signing Gonzalez would add to an already productive offseason for the Twins after picking up sluggers Nelson Cruz, C.J. Cron, and Jonathan Schoop to bolster a young core that includes Miguel Sano, Eddie Rosario, and Byron Buxton.
Gonzalez slashed .247/.324/.409 with 16 home runs in 145 games in 2018, and he's just one year removed from compiling a .907 OPS, along with 23 homers and 90 RBIs over 134 games for the Astros.
Minnesota is looking to bounce back after a lost 2018 season. The Twins finished with a 78-84 record after making it to the AL Wild Card game in 2017.