Citing a source it did not name, the newspaper said the $8.5 million guaranteed deal will hinge on Madson, 31, passing a physical.
The free-agent pitcher agreed in November to 4-year, $44 million deal to return to Philadelphia but the agreement was never consummated.
Madson's contract with Cincinnati includes an option year, the newspaper said. The Reds will not have to compensate Philadelphia with a draft choice due to new rules in baseball's recent collective bargaining agreement.
Madson racked up 32 saves in 35 chances last season, compiling a 4-2 mark with 2.37 ERA over 60 2/3 innings.