The mayor of Barcelona, Ada Colau, and the mayor of Valencia, Joan Ribó, had previously offered to take in the 629 migrants who are now on the Aquarius, after having been rescued from the Mediterranean by Doctors Without Borders (MSF) and Sos Mediterranée.
More than 600,000 mostly African migrants have reached Italy by boat from Libya in the past five years.
Following up on his campaign promises, the new Italian interior minister and deputy Prime Minister Matteo Salvini, leader of the right-wing League party, said the ship would not be allowed to dock in Italy and suggested that as it was closer to Malta, it should dock there.
By not letting migrants in, Mr Salvini is trying to avoid Italy to take upon itself this burden.
"In the Mediterranean.there's Malta that does now welcome anyone, there is France that rejects [refugees] at its borders, there's Spain that defends its borders with the arms. This time we say no", Mr Salvini said, urging Malta, to accept the ship. It said in a statement that the Aquarius took on the passengers in waters controlled by Libya and where Italian authorities in Rome coordinate search-and-rescue operations.
By law, it would be hard for Italy to refuse the boat a safe haven, as its own coastguard coordinated the rescues, picking up more than 280 migrants in its own vessels before transferring them to the Aquarius to be taken to safety.
Malta refused, saying it had nothing to do with the rescue mission, which was overseen by the Italian coastguard.
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There are no showers or sinks for migrants on the Aquarius and rescue workers use a disinfectant spray on people's hands before they eat.
Speaking to news channel Sky TG24, Aquarius crew member Alessandro Porro said that they were north of Malta and had "received no communications from the Maltese authorities".
The migrant influx has led to a populist backlash, with Salvini vowing to prevent Italy from becoming the "refugee camp of Europe".
"We call on all involved to contribute to a swift resolution so that the people on board the Aquarius vessel may be safely disembarked as soon as possible", European Commission spokesman Margaritis Schinas told CNN Monday.
MSF Sea (Doctors Without Borders) said the medical situation of those on board was "stable for now but unnecessary delay to disembarkation in safe port puts vulnerable patients at risk".
But the Spanish ports may be too far away for the overburdened Aquarius. On Sunday evening, he posted an image of himself on Twitter with his arms crossed above the message, "We are closing the ports", rendered as an Italian hashtag.
Italy has had to accept some 700,000 migrants since 2013, in line with the controversial Dublin Regulation requiring to declare refugee status in their first country of arrival.
The migrants were rescued in six separate nighttime operations in the central Mediterranean on Saturday.