Team president Bruce Allen said the team is working to re-sign Johnson, JP Finlay of NBC Sports Washington reports.
Washington signed Johnson on Dec. 4 after McCoy was injured. He backed up Mark Sanchez for one game before starting the final three games.
Johnson, 32, went 1-2 as a starter and completed 57.1 percent of his passes for 590 yards with three touchdowns and four interceptions last season.
He has appeared in 33 games with eight starts in his career.
Johnson originally was slated to play in the Alliance of American Football this season after the San Diego Fleet made him the first pick. But the AAF released Johnson from his contract to return to the NFL.