The vessel, operated by the French-German humanitarian group SOS Méditerranée and Médecins Sans Frontières, was carrying 600 migrants also rescued off the coast of Libya.
A day after the French president, Emmanuel Macron, said Rome had acted with "cynicism and irresponsibility" in turning away a migrant rescue ship, Italy's hardline new interior minister, Matteo Salvini, said it had "nothing to learn from anyone about generosity, voluntarism, welcoming, and solidarity".
But earlier, Italy's foreign ministry warned that relations had been compromised by France's public criticism that Italy's decision to refuse entry to the migrant ship Aquarius had been "cynical" and irresponsible.
Italian Prime Minister Giuseppe Conte on Thursday confirmed his planned Friday meeting with French President Emmanuel Macron, after a conciliatory phone call between the two politicians.
"France does not want escalation; it's counter productive. If we get one, we can start down a new path", Di Maio said in a radio interview. That could take up to two years, the source said.
Conte's office also issued a statement echoing the French account and confirming the meeting.
Salvini has vowed to stop foreign boats carrying migrants from arriving in Italy, a policy that has also affected a USA military ship.
A European Union summit later in June is due to consider changes to asylum law to better share the burden of incoming migrants.
In a swipe at France, Italy's government said Tuesday it would not accept "hypocritical lessons from countries that have preferred to look the other way on immigration".
Spain's new Socialist government has since agreed to allow the Aquarius and its 629 migrants to dock in Valencia.More news: Donald Trump, Kim Jong Un sign 'comprehensive document' after historic summit
Recall, the ship Aquarius with 629 illegal migrants refused to accept the ports of Italy and Malta.
The Aquarius remains at sea south of Sicily, and will land in Valencia around Saturday, Doctors Without Borders said.
Ropes have been strung around the ship to make it easier to walk on deck, video from the Aquarius showed.
In the Hungarian case, 59 men, eight women and four children from Syria, Iraq and Afghanistan suffocated in the back of a refrigerated truck with Hungarian license plates.
Despite criticism from the pope and the United Nations refugee agency, which called the treatment of the migrants on the Aquarius "shameful", Salvini has not changed his position.
The NGO said on Twitter the move was to "protect itself from the otherwise unbearable bad weather for the people on board, who are exhausted, suffering from shock and seasickness".
"None of the comments by French authorities have questioned this, nor the need for a close coordination between Europeans", a statement said.
Search-and-rescue ship who are "stuck" in the Mediterranean sea from Sunday, began their long journey to Spain on Tuesday evening after more than 500 migrants rescued by the coast guard of Italy.
Still at sea were 40 migrants rescued by the U.S. Navy vessel Trenton, which was in the Mediterranean for what the U.S. 6th Fleet said were routine operations.