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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Norwich City and Middlesbrough go head-to-head this evening at Carrow Road in a pivotal clash in The Championship promotion race.
Ipswich is a seat that has swung between the Conservatives and Labour for much of the last century.
A dry weekend's expected everywhere with plenty of sunshine, especially on Saturday.