The migrants, including pregnant women and scores of children, were saved by the French charity SOS Mediterranee on Saturday.
The organizations said the boat was located 35 nautical miles from Italy and 27 nautical miles from Malta.
The migrants were rescued in an operation coordinated by Rome Rescue Coordination Centre (RCC) within Libyan territorial waters, in an area that is closer to Tunis and Lampedusa than it is to Malta.
The decision was backed by other mayors in Italy's south, including those in Naples, Messina and Reggio Calabria, who all said they were ready to disobey Salvini's order and allow Aquarius to dock and disembark in their seaports, the Guardian reported.
Salvini, the leader of the far-right League, was rejected by Malta who said the ship fell under Italy's jurisdiction. "People are in distress, are running out of provisions and need help quickly".
The EU and the bloc's biggest member state Germany made similar pleas. In practical terms, no ships of this type will be able to land without cooperation of the Italian coastguard, which is under the direct command of the Italian government.
The Maltese government, however, was not moved. If they're going to refuse to accept refugees picked up at sea, what basis do they have to criticize Italy for balking at taking in hundreds more of them every day? "This time we say no".
The Aquarius and its passengers were caught up in a crackdown swiftly implemented by the right-wing partner in Italy's new populist government, which has vowed to stop the country from becoming the "refugee camp of Europe".
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The MV Aquarius as migrants and refugees are rescued by the SOS Mediterranee organisation in the Mediterranean Sea, off the Libyan Coast, September 14, 2017. "The details of this agreement are not known and we don't know if Salvini will inherit that deal", he said. It said just under 70% of the 171,635 migrants who entered Europe by sea had arrived in Italy.
This is something of a PR disaster for the new government, particularly since the shipload of refugees includes seven pregnant women and dozens of unaccompanied minors.
"Malta is respecting global rules and, therefore, can't let Aquarius dock at its port". "The people we saved were in a hard condition, at least 50 were at risk of drowning", he added.
He tweeted: #Chiudiamoiporti. "We're closing the ports".
"Hundreds of people urgently need assistance, slowing down operations puts their well being at risk".
Some in Italy have offered to take in migrants.
Barcelona's mayor Ada Colau also offered to welcome the ship, arguing that saving human lives should be prioritised.
"Our land has always been one of refuge, I don't know how you can send away 629 human lives".
"Italy has stopped bowing its head and obeying, this time THERE IS SOMEONE WHO SAYS NO. #chiudiamoiporti (#weareclosingtheports)", Salvini wrote on Twitter, as quoted by the Ansa news agency.
European Union leaders in December had set an end-June deadline for an overhaul of rules to create a permanent mechanism to deal with migrants in the event of a new emergency.