The person spoke on condition of anonymity Saturday night because there was no completed contract.
Greinke is the top arm on the open market, and has attracted interest from the Dodgers, Texas and the Los Angeles Angels. The 29-year-old righty started last season with Milwaukee and was later traded to the Angels, going a combined 15-5 with a 3.48 ERA.
The 2009 AL Cy Young winner with Kansas City would be the latest high-priced player to be acquired by the Dodgers. They traded for first baseman Adrian Gonzalez, shortstop Hanley Ramirez, pitcher Josh Beckett and outfielder Carl Crawford late last season.
With the Dodgers, Greinke would pair with 2011 NL Cy Young winner Clayton Kershaw for a formidable top of the rotation. The Dodgers, infused with money after the sale this year to an ownership group that includes Magic Johnson, went 86-76 last season and finished eight games behind eventual World Series champion San Francisco in the NL West.
Greinke is 91-78 with a 3.77 ERA in nine seasons with the Brewers, Dodgers and Angels.
Greinke and slugger Josh Hamilton were the top two free-agent targets at the baseball winter meetings this week. Texas was interested in re-signing Hamilton, and he also drew attention from Seattle.
Several potential trades and signings throughout the majors were thought to be on hold while Greinke and Hamilton made their decisions.