Posh will take on Chesterfield in the JPT final at the end of March, but Washington believes that his team could return a couple of months later in the League One play-off final.
"Obviously, the place is buzzing with the lads getting to Wembley," Washington told ITV News Anglia.
"Everyone's really excited and it's a great opportunity for all the lads. Now we've got to fight hard for the rest of the season to hopefully get there again."
Peterborough travel to local rivals Stevenage on Saturday in a game which carries huge significance at both ends of the table.
More top news
Northamptonshire and Norfolk are experiencing a 'mini-boom' in house price growth according to property website Rightmove.
Patchy rain in south overnight. Occasionally bright on Monday.
Police patrols are continuing in Cromer. Businesses say they are open as usual but there were reports of fresh trouble in the town earlier