Madson saved 32 games in 34 chances last year for NL East champion Philadelphia. CBSSports.com was the first to report the right-hander's agreement with Cincinnati.
The contract is subject to a written agreement and a physical, according to the person who spoke Tuesday night to The Associated Press on condition of anonymity because the deal had not been finalized.
Madson replaces Francisco Cordero as the closer in Cincinnati. Cordero remains a free agent after the Reds declined a $12 million option, choosing instead to pay him a $1 million buyout.
Madson nearly re-signed with the Phillies, but a $44 million, four-year deal collapsed in November and days later Philadelphia reached a $50 million, four-year contract with closer Jonathan Papelbon.
In 491 career games, including 18 starts, Madson is 47-30 with a 3.59 ERA and 52 saves.
After years of success in a setup role, he took over for an injured Brad Lidge as Philadelphia's closer last season and went 4-2 with a 2.37 ERA in 62 games. The lanky Madson allowed only two home runs and 16 walks in 60 2-3 innings.
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.
AP Sports Writers Mike Fitzpatrick and Jay Cohen contributed to this report.