A few days ago the Rossoneri were ready to hand over the Brazilian defender for 47m, but pulled the plug at the last minute on the deal with PSG.
"It was a big decision and President Silvio Berlusconi gave a huge gift to the team and fans," Allegri told Sky Sport Italia.
"Ibrahimovic will also stay. He had a strong season, scoring 28 goals. The club decided not to dismantle the squad and will keep the foundations of the team intact.
"The side is almost complete, but the market is open until August 31 and anything can still happen."