The deal is for four years, $56 million.
A news conference is scheduled for 3 p.m. ET.
A baseball source said the Dunn signing does not preclude the White Sox from trying to re-sign Paul Konerko and that Dunn was attracted to the team in part by the possibility of playing in the same lineup with the veteran first baseman.
Chicago general manager Kenny Williams pursued Dunn at the trade deadline in July. The team also spoke to Dunn's representatives at the recent general manager meetings, sources told ESPNChicago.com's Bruce Levine.
Although Dunn has expressed an interest in staying in the National League and continuing to play first base, most scouts consider him better suited to the designated hitter role at this stage of his career.
The designated hitter spot was a major concern for the White Sox last season. Manager Ozzie Guillen's DH contingent hit only 18 home runs and logged a .728 aggregate OPS.
Jerry Crasnick is a senior writer for ESPN.com. Bruce Levine covers baseball for ESPNChicago.com. ESPNChicago.com's Doug Padilla and The Associated Press contributed to this report.