"European governments have put all their efforts into propping up the Libyan JRCC, however, Friday's events show that they do not have the capacity to fully coordinate a rescue", said Aloys Vimard, MSF's Project Coordinator onboard Aquarius.
Matteo Salvini, the country's far-right populist interior minister, also wrote: "It can go anywhere it wants, NOT in Italy".
The European Commission was in touch with several EU states and trying to help resolve the "incident" with the Aquarius, a spokesman in Brussels said.
A United Kingdom government spokesperson said: "We are deeply concerned for the welfare of 141 migrants who have been rescued from the Mediterranean by sailing vessel Aquarius".
More than 650,000 migrants have come to Italy's shores since 2014, but the numbers of new arrivals have plunged over the past year, with Rome encouraging the Libyan coastguard to carry out most of the rescues. She said "it could of course be the case that in theory a flag state of a rescue ship could be considered a potential location for disembarkation, but this might not be possible in practice". It is believed they had been at sea for almost 35 hours.
Malta said it had no legal obligation to berth the ship and Spain said its ports were not the safest destination.
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The group said many reported they had been held in inhumane conditions in Libya. It said more than 70 percent are from Somalia and Eritrea, whose citizens are often eligible for refugee status. But on Monday, Madrid did not repeat its offer, with a government spokeswoman saying: "At the moment, Spain is not the safest port because it is not the nearest one" for Aquarius to dock. "Ships might be unwilling to respond to those in distress due to the high risk of being stranded and denied a place of safety", said Vimard.
The group was picked up in the Mediterranean Sea by French humanitarian workers on Friday.
"What is of utmost importance is that the survivors are brought to a place of safety without delay, where their basic needs can be met and where they can be protected from abuse", said the group's search and rescue coordinator, Nick Romaniuk.
More than 650,000 migrants have come to Italy's shores since 2014, and Rome has accused its European Union peers of not sharing the burden of taking in those who arrive on the bloc's southern border. Malta says it can't handle large numbers of migrants.
The migrants were rescued last week after their two wooden boats capsized.
According to the statement, Aquarius was made off the Libyan coast, Tunisia and Italy (Lampedusa) in an operation which was coordinated by the Libyan authorities.