The deal, pending completion of a physical, would pay Willis about $1 million and include additional money for performance bonuses, the source said.
He held lefty hitters to a .127 average (7-for-55) with the Reds last season, and had attracted interest from Seattle, Toronto, Boston and Cincinnati this winter as a left-handed arm out of the bullpen.
Willis won the 2003 National League Rookie of the Year award with Florida. His best season came in 2005, when he went 22-10 with a 2.63 ERA for the Marlins and finished second to St. Louis' Chris Carpenter in Cy Young Award balloting.
Jerry Crasnick is a senior baseball writer for ESPN.com.