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
Generally cloudy with occasional outbreaks of rain during the morning. Some glimpses of the sun are possible. Feeling mild
We're one week into December and the countdown to Christmas is well underway. Read the latest in ITV News Anglia's Christmas Shout Outs.
The grandparents of missing serviceman Corrie McKeague have travelled 425 miles from their home in Scotland to retrace his last steps.