The deal, signed in November by Italian Prime Minister Giorgia Meloni and Albanian counterpart Edi Rama, was sharply criticised by human rights groups.
They said the deal is illegal under international law, warning that Albania, a non-EU country, offers limited protection for asylum seekers.
"Everything is ready to take in migrants" in two holding centres, Fabrizio Bucci told AFP during a visit to the facilities.
"As of today, all the structures are functional," he added.
The migrants deemed most vulnerable – including women and children – are set to be taken to Italy, with the rest set to be sent to the centre in the northern Albanian port of Shengjin.
After registering there, they will be taken to another centre at a nearby former military base in Gjader, while they wait for their claims to be processed.
Around 10 Italian judges will oversee hearings with asylum seekers in Albania, according to Italian media reports.
In the Italy-run camps, the migrants will be accommodated in prefabricated buildings of around 12 square metres, surrounded by high walls and guarded by police.
Over 300 Italian soldiers, doctors and judges are involved in the operation, the ambassador said.
Italy and Hungary have proposed extending the system across the EU by creating "hubs" at the bloc's external borders where people with no right to stay in Europe could be housed while deportation is arranged.
The idea could be on the agenda of an EU summit set to be held in Brussels next week.
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