The Spanish prime minister Pedro Sanchez said he would give the ship "safe harbour" and that he wanted to help avoid a humanitarian emergency.
Médecins Sans Frontières (MSF), also known as Doctors Without Borders, which is operating the Aquarius alongside SOS Mediterranne, urged a rethink.
French President Emmanuel Macron, who had been criticized for not offering to take in the boat at a French port, broke his silence on the topic during a Cabinet meeting Tuesday, denouncing the "cynicism and irresponsibility of the Italian government", government spokesperson Benjamin Griveaux said at a press conference.
Doctors Without Borders, which has staff on board, said 15 of the migrants had significant chemical burns - common among those who sit on the floor of smugglers' boats, where gasoline mixes with seawater.
File Image of MV Aquarius, a search and rescue ship.
The EU and the bloc's biggest member state Germany made similar pleas.
A rescue ship carrying more than 600 migrants will be allowed to enter the Spanish port of Valencia, after it was turned away from ports in Italy and Malta.
But the new Italian government, keen to show its racism is honest, would not allow the ship to dock. Salvini retweeted a video with Strada's comments, adding "Stop human trafficking, let's defend our boundaries, I won't give up, my friends".
The standoff marked the first inevitable clash over migrant rescues with League leader Salvini, now running Italy's interior ministry.
The migrant influx has led to a populist backlash, with Salvini vowing to prevent Italy from becoming the "refugee camp of Europe". "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.
Muscat said he told Conte "that Malta is acting in full conformity with its global obligations".
People on board the vessel, which has been stranded in the Mediterranean after both Italy and Malta refused to allow it to dock at the weekend, will be transferred to Italian ships and be given food and water. A further 41 people, it said, had been rescued by a US Navy vessel.
UNHCR spokesman Andrej Mahecic says rescue at sea is a long tradition, upheld by International Maritime Conventions.
Late on Monday, the Aquarius's rescue coordinator, Nicola Stalla, had said that it would be impossible for the entire group of migrants to make the journey to Valencia on board the rescue ship, as it would mean sleeping out in the open in deteriorating weather conditions.
"MSF calls for people's safety to come before politics".
It took six different operations by a French charity called SOS Mediterranee to pick up all 629 migrants and get them aboard a rescue ship called Aquarius.
Spain has accepted a rescue ship carrying 629 migrants which had been turned away by the new Italian government, avoiding a humanitarian crisis and diffusing a political stand-off between two European Union countries. The mayor of Taranto, Rinaldo Melucci said the southern port city was "ready to embrace every life in danger".
The summer of 2016 saw a surge in mass drownings in the Mediterranean, at the peak of Europe's worst migration crisis since World War II.