The 31-year-old right-handed reliever is married to Desmond's sister Nikki.
The Nationals attended the 33-year-old right-hander's workout in Phoenix. Madson hasn't pitched in the majors since 2011 because of Tommy John surgery.
The Nationals missed out on the power-hitting right-handed versatile infielder but remain interested in the left-handed reliever.
"We like our bullpen. We like our core players. We certainly like the construction of this roster."
The veteran right-handed closer who drew the Nationals' interest is reportedly signing with Tampa Bay.
The Nationals talked to several free agent left-handed relievers, made an offer to one and explore as many as four trade options.
The new manager is a proponent of two left-handers in the bullpen, a specialist and a long man.
The Nationals general manager says the interest in both relievers has been informal.
Mike Rizzo says the team's goals now are to upgrade the bench and bullpen.
The 22-year-old left-handed reliever has a 6.75 ERA since the all-star break.