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
A woman has been banned from keeping all animals for three years after admitting neglecting her cat and four kittens - two of whom died.
With the cold weather expected to last for at least the next five days, the Met Office has issued various snow and ice warnings.
Cold but dry and bright this afternoon followed by a risk of icy stretches tonight.