Citing team sources, the outlets said Soriano has inked a contract worth $28 million covering the 2013 and 2014 seasons.
MLB.com said the deal also includes a $14 million option for 2015 that vests if Soriano finishes 120 games over the next two years.
The 33-year-old right-hander is coming off his best season in 2012, when he took over the closer role for the New York Yankees from an injured Mariano Rivera and responded by posting a 2.26 ERA with 42 saves.
The Post reported Nationals owner Ted Lerner was personally involved in the negotiations to land the top-flight reliever after Washington's bullpen faltered during the 2012 National League Division Series against St. Louis.
The Nats surrendered 16 earned runs in the series, including a meltdown in the decisive Game 5 when the bullpen blew a 7-5 lead going into the top of the ninth inning and ended up losing 9-7.