The 36-year-old Italy midfielder was in tears at the end of the 3-1 defeat to Barcelona in the Champions League final, amid speculation he is set to end his 20-year Serie A career and move to the MLS.
But while his manager was keen to talk up the progress of his side - who sealed an Italian league and cup double this season - he refused to say whether Pirlo will stay in Turin,
"As far as Pirlo is concerned, I can't give you an answer I'm afraid," Allegri said.
"As for Juventus, I think we can improve still. Not necessarily results because that would mean winning all three trophies.
"What we can improve is the quality of our game. We can consolidate and strengthen our way of playing and stay among the top eight teams in Europe on a regular basis.
"This is the most important objective for the club."
Juventus fell behind early on in Berlin to an Ivan Rakitic goal, but stunned Barcelona when Alvaro Morata equalised 10 minutes after half-time.
However, Juventus' hopes of upsetting the odds were destroyed when Luis Suarez fired home the decisive goal after good work from Lionel Messi before Neymar sealed Barcelona's treble-winning season with the final kick of the match.
And speaking at his post-match press conference, an optimistic Allegri was left to rue the "minor details" that cost his side a treble of their own.
"We didn't manage to equalise. That's what was missing," he joked.
"They did well to score with the first chance they had. From the time we equalised to the time they scored the second goal, we controlled the game but Messi made a wonderful move, Buffon made a wonderful save and Suarez hit the net and managed to bag a goal.
"During a final, you have to be very careful because the thing that tips the balance is the minor details."