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.
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After retiring from county cricket last year, Graham Napier has embraced a new challenge: taking up a top coaching role in schools cricket.
After a chilly start it will be dry with increasing amounts of sunshine later.
Chilly, with clear skies overnight. Sunny Saturday, and warm inland.