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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A campaign has been launched to improve one of the key commuter routes in the East, linking parts of Essex and Cambridge to London.
Former Pink Floyd frontman Syd Barrett is being a remembered at a special concert at the Cambridge venue where he played his last gig.
The rather cloudy skies will remain over the next few days but it will stay mostly dry. Some fog overnight at the weekend.