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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It will be a largely cloudy but humid day with the risk of hit-and-miss heavy showers and thunderstorms.
It will turn dull and misty again overnight but it will stay mild. Tomorrow will be mostly cloudy with a risk of localised heavy showers.
Group Lotus has announced plans to shed over 300 jobs in a restructuring and cost-cutting move.