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
Cladding from high-rise flats in 25 local authority areas has now proved to be potentially flammable in the urgent safety review.
Scattered showers at first this evening will gradually decay. A dry night to follow with clear spells developing.
A willow whale is taking shape on the beach in Harwich as part of the town's arts festival