Montolivo is joining on a Bosman from Fiorentina.
"I must compliment Juventus, as they had an extraordinary year and deserve this Scudetto. This does not mean it was a failure for Milan, though," said Allegri.
"We won it last season, went close this time and hope to be competitive in Europe and Serie A next year too.
"Next season Montolivo will arrive and he is a very talented player with great technique. In any case, Milan will remain competitive next year and we must continue the work we started last season. We need to begin again with great enthusiasm."