French President Emmanuel Macron accused Italy's new populist government of "cynicism and irresponsibility" for closing its ports to the 629 migrants.
The French president's office has confirmed the visit to France of Italian Premier Giuseppe Conte to meet with French President Emmanuel Macron on Friday.
The vessel, run by charity organizations, was denied port entry also by Malta, and was eventually offered to dock in Spain.
But Italy's foreign minister, Enzo Moavero, kept up the barrage by telling a French envoy that Macron's comments were "unjustifiable" and had compromised relations between the two countries. The French president's office said France didn't want to "start a precedent" that would allow some European countries to breach global laws and rely on other EU member states to take in migrants.
While acknowledging that the controversial European Union quota system - which would have seen illegal immigrants arriving to the bloc forced on unwilling nations - was "a failure", the UN figure insisted: "Unless Europe goes back to this notion of sharing the burden, we will not get out of this situation".More news: Monaco to sell Lemar to Atletico Madrid
The saga began on Monday when Italy refused to let the Aquarius dock and disembark passengers in Italy, insisting Malta should do so.
Foreign Minister Enzo Moavero Milanesi asked to see the diplomat "following the statements made yesterday in Paris", a ministry statement said on Wednesday.
Kurz said Europe needed an "axis of the willing in the fight against illegal migration".
Interior Minister Matteo Salvini was to brief Parliament later Wednesday on the situation as the Aquarius and its 629 passengers continued its dayslong, westward voyage to Spain.
Meanwhile, an Italian coastguard ship carrying more than 900 migrants was allowed to dock in Sicily on Wednesday.More news: North Korea Summit: Trump and Kim pledge to leave the past behind
Salvini has accused European aid groups of essentially operating taxi services for Libya-based human traffickers, and has said Italy will now refuse them entry.
It is now en route to Valencia, escorted by Italian navy and coast guard units.
France, she said, had also taken the second-biggest share of migrants relocated from Greece and Italy.
"We need to listen" to Italy, Grandi said, stating that the nation's populist government was "right to say" the nation can not welcome every boat of third world migrants wishing to make Europe their home.
The number of migrants arriving in Italy began plummeting previous year after the Italian government under the centre-left Democratic Party negotiated controversial deals with Libya that beefed up its capacity to better patrol its coasts and discouraged land-based smugglers.More news: Why Meghan Markle stood behind Kate Middleton for her balcony debut
The U.N. refugee agency has said that approximately 1,900 migrants came to Cyprus in about 40 boat trips since 2015.