USA Today earlier reported the deal, which the White Sox have yet to confirm.
It would be the second free agent the White Sox have added this week. They brought on left-handed reliever Zach Duke for bullpen depth Tuesday.
A .264 hitter over 11 seasons, the 35-year-old LaRoche has 243 career home runs. He hit 26 of them this past season for the Washington Nationals to go along with 92 RBIs.
"It's on the list of needs," Hahn said Tuesday. "Obviously we want to have the balance in the lineup and allow [manager] Robin [Ventura] the flexibility to play the right matchups when the opportunity arises. Right now we do tilt a little heavily right-handed, especially in the middle of the order."
LaRoche has played all but six of his nearly 1,500 career games in the National League. He played those six games with the Boston Red Sox in 2009.
He finished sixth in the NL MVP race in 2012, hitting 33 home runs with 100 RBIs for the Nationals. He was also a Gold Glove Award and Silver Slugger award winner at first base that year.
LaRoche figures to pick up the bulk of his at-bats as the White Sox's designated hitter, essentially taking over the job that had been Adam Dunn's for the past four years. He also figures to spell Abreu at first base on occasion.
ESPN.com's Jerry Crasnick and Jim Bowden and ESPNChicago.com's Doug Padilla contributed to this report.