Joe Mauer, who announced his retirement on Nov. 12, was the Twins' primary first baseman for the past five years.
Minnesota first basemen combined to post a .711 OPS, which ranked 26th in Major League Baseball last season.
ESPN's Buster Olney reported last month that Arizona was willing to listen to offers for its best players after a disappointing finish to the 2018 season.
The Diamondbacks spent 125 days leading the National League West, but an 8-19 record in September kept them out of the playoffs for the sixth time in the past seven seasons. They already have $77.5 million committed to seven players in 2019.
Goldschmidt, 31, will make $14.5 million in the final season of his contract. He has made the NL All-Star team in each of the past six seasons and ranks third among all MLB players with 32.9 FanGraphs wins above replacement since 2013.
During the 2018 season, Goldschmidt posted a .290/.389/.533 slash line with 33 homers in 158 games.