Spanish newly sworn-in Prime Minister Pedro Sanchez said he instructed the ship to be admitted to the eastern port of Valencia.
Doctors Without Borders (MSF), which is operating the Aquarius alongside SOS Mediterranne, urged a rethink.
The Spanish government said it would accept the ship at the port of Valencia.
The Aquarius was denied the right to dock in Italy and Malta.
French charity Medecins Sans Frontieres said the European response to accepting hundreds of refugees trapped in Meditarranean limbo over the weekend was "not a victory". But he quoted Macron as telling Tuesday's weekly Cabinet meeting: "If any ship was closer to France's shores, it could obviously dock on the French coast".
The EU and the bloc's biggest member state Germany made similar pleas.
The head of the French charity which charters the Aquarius migrant rescue ship said Tuesday that it would continue its operations despite the worldwide standoff over the 629 people now onboard.
French President Emmanuel Macron criticized what he called Italy's cynicism and irresponsibility for leaving the migrants at sea, while also deflecting criticism for not allowing the ship to dock in France. It's an argument that human rights advocates also have supported.
SNP's Ian Blackford clashes with Speaker before walkout
The Scottish First Minister added: "The decision to act without our consent, and the manner of doing it, will not be forgotten". Such a complex endeavour is bound to complicate the agenda of Prime Minister Theresa May as she tries to negotiate Brexit .
It's not yet clear what Salvini and the new government have in mind, but it's worth remembering that the League's Roberto Maroni was interior minister in 2009 when Italy implemented its nefarious policy of literally pushing boats back to Libya.
The migrant influx has led to a populist backlash, with Salvini vowing to prevent Italy from becoming the "refugee camp of Europe". "It is our obligation to avoid a human catastrophe and offer a safe haven to these people, fulfilling that way the obligations of global law", his government said in a statement.
SOS Mediterranee tweeted late on Sunday that the Aquarius had "received instructions from the Italian Maritime Rescue Coordination Centre to standby in our current position which is 35 nautical miles from #Italy and 27 nautical miles from #Malta".
"Our mission is to rescue people at sea", SOS Mediterranee general manager Sophie Beau told AFP.
A U.S. Navy ship, Trenton, radioed Sea Watch on Tuesday to say it had picked up 12 bodies and 41 survivors from a sinking rubber boat. The passengers were rescued in seven separate operations. "(We) call for people's safety to come before politics".
But the decision to close ports was significant because it showed Italy's willingness to put up a stop sign closer to its border.
But Italy can not simply expel hundreds of thousands of undocumented migrants without severe repression, and Italy doesn't yet have the agreements with many key countries of origin needed to facilitate returns.
"Naples is ready, without funds, to save lives", he said.
"Saving lives should always be our top concern".
European Union leaders in December had set an end-of-June deadline for an overhaul of rules to create a permanent mechanism to deal with migrants in the event of a new emergency.
Mediterranean since 2013, often from war-wracked Libya.