NFL Network was first to report the deal.
McCoy is now in prime position to back up Jones, the second-year quarterback who was the No. 6 overall pick last year. He served as a positive mentor last season for rookie Dwayne Haskins in Washington. He also previously worked well alongside a young Robert Griffin III.
Tanney, 32, is a holdover from the previous coaching regime who had his contract option picked up several weeks ago. Tanney's deal does not include any guaranteed money. He served as the third-string quarterback last year behind Jones and the now-retired Eli Manning.
McCoy has started 28 games in his career, including Week 5 this past season against the New England Patriots. But he has an extensive injury history.
McCoy, subbing for the injured Alex Smith, broke his right leg in a Dec. 3, 2018, loss at Philadelphia and tried to make a quick return in case the Redskins reached the postseason. But McCoy was never ready and endured complications in his recovery. He needed three surgeries on his leg in the offseason, causing him to miss all of the spring workouts. He returned for training camp, only to suffer more problems because his right leg was not strong enough to plant and move.
McCoy did make one start, a Week 5 loss to New England that was also Jay Gruden's last game as a coach. McCoy remained healthy but did not play again the rest of the season.
The former University of Texas star has appeared in 40 career games. He has completed 60.8% of his passes for 6,080 yards with 29 touchdowns and 27 interceptions.
ESPN's John Keim contributed to this report.