The newspaper said Berisha's home in the village of Prifc near Tropoja, in northern Albania, was the center for the alleged smuggling operations.
Berisha was then an opposition politician and former Albanian president. The report said hundreds of thousands of weapons and ammunition went missing during the period.
Politika said four ethnic Albanians from Kosovo named Berisha as a smuggler in an investigation into Serbian war crimes.
Most of the arms ended up in the hands of the Kosovo Liberation Army, Politika said. It said the four witnesses are former members of the KLA arrested by Serbian security forces during the war.
The war in Kosovo started in the spring of 1999. A recent report claimed the KLA kidnapped Serbs and other civilians, imprisoned them, and then had their organs removed to sell on the black market.