Both players received an invitation to spring training.
The six-time All-Star has undergone two Tommy John surgeries in his career.
Albers, 34, was 2-6 with a 6.31 ERA in 59 appearances for the Chicago White Sox last season.
Shawn Kelley is expected to be the closer for the Nationals this season after Mark Melancon signed with the Giants in free agency. The Nationals also made a bid to sign Kenley Jansen, who re-signed with the Los Angeles Dodgers.
In addition to Nathan and Albers, the Nationals also announced that they signed Vance Worley to a minor league deal with a spring invite on Tuesday. Worley, 29, posted a 3.53 ERA in 35 games (four starts) for the Baltimore Orioles last season.