About 72,000 migrants arrived in Italy on the perilous route from Libya between January 1 and June 21, roughly 20 percent more than in 2016, and more than 2,000 died on the way, according to the International Organization for Migration.
Italian news agency ANSA said Italy was proposing a ban on non-Italian-flagged rescue ships disembarking migrants in Italian ports, though not those participating in the EU's Frontex rescue operation.
More than 10,000 asylum seekers arrived in the country between Saturday and Tuesday, and another 12,000 have arrived in the past 48 hours.
The Italian government is considering blocking boats carrying migrants from landing at its ports after almost 11,000 refugees arrived on its shores in five days.
Most of the people who are rescued in the Mediterranean are those who try attempting the crossing from crisis-hit Libya as it is a gateway to Europe for migrants from across sub-Saharan Africa, the Arabian Peninsula, Egypt, Syria and Bangladesh. All of those rescued off the coast of Libya are brought to Italy, often by private charities.
Among the NGOs rescuing migrants in the Mediterranean is Doctors without Borders (MSF).
"We are now facing an biblical exodus", said Paolo Romani, head of ex-Premier Silvio Berlusconi's opposition Forza Italia party in the upper house of Parliament.
Traffickers on the North African coast take advantage of periods of good weather to set large numbers of migrants seeking passage to Europe out to sea, a notoriously risky crossing.
Avramopoulos said, "Italy is right that the situation is untenable", adding that the country's management of the crisis was "exemplary".
Italy has threatened to stop foreign boats carrying migrants to its ports. And even the maverick radical Five Star Movement is moving to a more anti-immigrant position, calling for a halt to any new migrants being lodged in Rome.
Avramopoulos responded via Twitter that he had discussed with the Italian envoy "our support after increasing numbers of arrivals in Italy".
Avramopoulos made nearly exactly the same remarks in February, and similar promises have been made by other European Union officials.