"We can't go on like this," the source told AFP after media reported that Italy's representative to the European Union, Maurizio Massari, delivered a letter to the bloc warning that the country was considering closing its doors.
Massari is said to have met the European Commissioner for Immigration, Dimitri Avramopoulos, to insist that the latest wave of people saved from rickety vessels in the Mediterranean had meant "the situation has become unsustainable".
Prime Minister Paolo Gentiloni said Italy was "asking certain European countries to stop looking the other way, because we cannot go on like this".
The threat follows a poor performance by the ruling centre-left Democratic Party (PD) at local elections this weekend, where it lost heavily to the centre-right Forza Italia and the anti-immigrant Northern League party.
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The anti-establishment Five Star Movement (M5S), which is running neck-and-neck in national polls with the PD, has followed the Northern League in demanding a stop to new arrivals -- particularly in the capital Rome.
"Denying docking rights to ships who have rescued people in distress at high seas (a legal obligation for all vessels, including commercial ships) sounds dubious.
"It sounds more like a desperate way to get Europe's attention than a real measure that can be implemented," he said.
More than 10,000 people were pulled to safety between Sunday and Tuesday off Libya -- including 5,000 on Monday alone.
While the Italian coastguard coordinates the operations, many ships sailing under foreign flags take part including several aid ships chartered by privately-funded organisations from France or Germany.
Critics have said the NGOs attract traffickers by sailing close to the Libyan coast. They insist they have no choice, because smugglers put the migrants out to sea in flimsy vessels that sink as they reach international waters.
The Maltese organisation Moas was the first to launch a private rescue vessel in the summer of 2014.
By last summer close to ten different NGOs, financed mainly through private donations, were taking part in migrant relief operations off Libya.