Both Italy and fellow European Union member Malta have refused to accept the migrants who are now heading for the Spanish port of Valencia after the new socialist government in Madrid agreed to take them in.
Italy on Tuesday threatened to cancel a summit with France after Paris strongly criticized an Italian decision to close its ports to a vessel carrying more than 600 migrants, according to Italian media.
The agency's special envoy for migrants in the Mediterranean, Vincent Cochetel, underscored that "people are in distress, are running out of provisions and need help quickly".
SOS Mediterranee director general Sophie Beau said Spain's offer "is encouraging and shows that some states are sensitive to the humanitarian emergency".
But the deteriorating weather and the extremely cramped conditions on board the Aquarius - which was built to transport a maximum of 500 - could make that journey to Valencia perilous. "The teams here onboard tell us that, for now, the situation onboard is under control".
The migrants will be taken by the Italian coast guard to Spain, but it is not yet clear if they will be taken there directly, or how the rest of the 629 will reach the Spanish mainland.
After rescuing 229 people from these boats, the Aquarius was requested by the IMRCC to accept a transfer of people who had been rescued by Italian navy and coastguard ships on 9 June.
The Aquarius itself would then "sail to Valencia to disembark the remaining 229 people", it added.
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"Denying disembarkation to desperate people rescued at sea can not be considered as a victory: it is the wrong response to the lack of responsibility and burden sharing between member states", said Aloys Vimard, MSF's project coordinator on board Aquarius.
Migrants wait to disembark from the rescue ship Aquarius in the Sicilian harbor of Catania, Italy, on May 27.
After nearly three months of jumping up and down in place, the newly invested Italian interior minister, Matteo Salvini, has hit the ground running in the worst possible direction.
"Italy is done bending over backwards and obeying - this time THERE IS SOMEONE WHO SAYS NO", he wrote on Twitter.
Anelise praised the teams of staff on the vessel: "The teams here are very calm, they say the situation is under control and we continue heading north towards the east part of Malta and towards Italy".
The migrants, including pregnant women and scores of children, were saved by the French charity SOS Mediterranee on Saturday.
Mr Salvini also took aim at other European countries that were involved in rescue operations in the Mediterranean but did not actually take in any migrants.
The outgoing Democratic Party government had already intensified cooperation with Libyan authorities to stop departures and enable Libyan coast guard forces to intercept migrant boats in global waters and take people back to horrific conditions in arbitrary detention in Libya.
The regional leader of Spain's Basque Country, Inigo Urkullu, said the region was prepared to take care of 10 percent of the passengers. It came just after a fatal day in the Mediterranean, with at least 112 people estimated drowned when a boat that departed from the Tunisian coast sank at sea, and nine people, including six children, died in the Aegean Sea between Turkey and Greece.