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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Dry and bright with some rain in the north and west, this clearing to drier weather for all for much of the day. Staying breezy.
Winds easing tonight and staying mostly dry. Some rain tomorrow with winds turning gusty again.
Hertfordshire police are now treating the case of a farmer who has been missing since June as murder. William Taylor was last seen in June.