The team is expected to open the season with Ivan Rodriguez and Wil Nieves behind the plate.
Coste, who will join the Nationals' big league camp, went 4-for-13 with one RBI in 10 Spring Training games with the Mets.
Coste, 37, appeared in 88 games last season with the Phillies and Astros, and hit .224 with two home runs and 18 RBIs.
The Nationals currently have 37 players in their big league camp.
Bill Ladson is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.