The Nationals had unconditionally released Norris last week. He at one point appeared to be penciled in as Washington's starting catcher but became expendable after the club signed free agent Matt Wieters.
A fourth-round draft pick by Washington in 2007, Norris was reacquired by the team in December via a trade with the San Diego Padres for minor league right-hander Pedro Avila. Norris batted .186 with 14 homers and 42 RBIs with San Diego last season.
With Washington's starting catcher in 2016, Wilson Ramos, gone after signing as a free agent with Tampa Bay, Norris was in line for the job. Until, that is, four-time All-Star Wieters joined the Nationals last month.
The Associated Press contributed to this report.