Construction at the centre at the Shengjin port, designed to accept migrants rescued at sea by Italy, has not been completed, port director Sander Marashi was cited as saying by Agence France Presse on Monday.
The opening was initially scheduled for 20 May.
The facilities are part of a controversial five-year agreement signed between the governments of Albania and Italy in November. One processing centre will be situated near the port and the other further inland, in the village of Gjader.
The deal has drawn criticism from opposition parties and human rights organisations, although it was endorsed by European Commission President Ursula von der Leyen.
The parliaments of both countries endorsed the agreement in February.
Italy will fully cover the costs of construction, security, and medical care for asylum seekers, with expenses estimated to reach up to a total 750 million euro, Italian media have reported.