The commission said final results will be available either late tonight or on 6 July.
Berisha already declared victory last night and his party held celebrations throughout the country.
Prime Minister Fatos Nano's Socialist Party has already begun complaining about alleged vote rigging.
With tension building, President Alfred Moisiu warned the commission to speed up the announcement of results, but the commission blamed the parties, saying their local officials were causing delays.
On 4 July, the Organization for Security and Cooperation in Europe (OSCE), in a joint statement with the Council of Europe and the European Parliament, said the poll failed to fully meet democratic norms.
Albanians Prepare To Elect A New Parliament