Madson, 31, was 47-30 with a 3.59 ERA in eight career seasons with the Philadelphia Phillies. He went 4-2 with a 2.37 ERA and 32 saves in 2011 after taking over for the injured Brad Lidge as Philadelphia's closer.
Madson's contract with the Reds includes a mutual option with a buyout, a source told ESPN.com's Jayson Stark. Although the Reds expect Madson to decline the option in order to pursue free agency again a year from now, the option and buyout will enable the Reds to push some of the money he's owed into next year.
Madson was reportedly on the verge of a multiyear deal with Philadelphia in November, but talks between Phillies general manager Ruben Amaro Jr. and agent Scott Boras mysteriously broke down late in the process and the Phillies signed Jonathan Papelbon to a four-year, $50 million deal.
Madson's efforts to land a long-term deal this winter were undercut by a surplus of other options on the market. Huston Street, Andrew Bailey and Sergio Santos are among the closers who have changed teams in trades, and Joe Nathan, Frank Francisco and Matt Capps all signed deals for one or two years in length.
Madson will slide into the Cincinnati closer's role previously held by Francisco Cordero, who saved 150 games for the Reds over the past four seasons. Cordero is still on the market as a free agent.
Cincinnati slipped to 79-83 last season, finishing third in the NL Central a year after winning the division. General manager Walt Jocketty has spent much of the offseason retooling his pitching staff, acquiring starter Mat Latos and left-handed reliever Sean Marshall in separate trades that cost the Reds pitchers Edinson Volquez, Travis Wood and several promising prospects.
Madson is coming off a three-year deal with the Phillies that guaranteed him $12 million. He earned $5.1 million last year, including $600,000 in bonuses for games finished.
Jerry Crasnick is a senior baseball writer for ESPN.com. Information from ESPN.com senior baseball writer Jayson Stark and The Associated Press was used in this report.