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
Canaries boss Alex Neil said the club are exploring a number of possibilities for a new left-back following the departure of Martin Olsson.
A Suffolk GP has written an open letter to the Prime Minister, inviting her to spend a day at his surgery.
The mother of serial killer Joanna Dennehy says she was once a little girl so caring she would cry if she accidentally killed a worm.