Sea Watch confirmed it had space and food for 41 passengers, but said it declined to take the migrants from the U.S. Navy ship because Italy had refused to assign the group's rescue vessel a port where it could dock.
The migrants aboard the Aquarius, run by French NGO SOS Mediterranee, were stuck on the overloaded ship as Italy and Malta bickered over who should take them.
A feud was triggered this week by Macron's tough assessment of Italy's decision to shut its ports to a rescue ship carrying 629 migrants - he called the Italian government's behavior "irresponsible".
An Italian coast guard ship carrying more than 900 Libyan migrants docked in Sicily Wednesday, just barely after the Rome government rejected entry to a boatload of others.
Many migrants in Ventimiglia have been "pushed" out of Italy's asylum system, which has failed to meet migrants' basic needs, the report says.
The Aquarius is expected to arrive at a Spanish port this weekend if weather permits.
The line Italy drew with the Aquarius appeared to be a tactic by the populist government sworn in this month to force Europe's hand at a summit of European Union leaders in Brussels set for June 28-29.
A French presidential source said the country had not received any formal demand from Italy for an apology.
The influx has encouraged the rise of far-right and populist parties - leading most recently to an anti-migrant coalition government taking power in Italy.
Malcolm Turnbull issues emphatic endorsement of Donald Trump's North Korea approach
But the statement did not specify his North Korean counterpart by name, saying it would be a "relevant, high-level" official. The North has long blamed the surrounding tense security environment on Washington's "hostile policy" toward it.
In addition to migrant children and families, Francis also said particular concern must be shown to those who are victims of human trafficking rings, and those displaced due to conflicts, natural disasters and persecution.
It says French police routinely stop unaccompanied children and put them on trains back to Italy after altering their paperwork to make them appear older, or to make it appear that they want to be sent back.
Earlier, Pope Francis said that the Gospel teaches that it's wrong to leave migrants "at the mercy of the waves", an apparent critique of Italy's new migration policies.
But hours later, a remorseful Macron called Conte to apologise, saying he had "no intention to offend Italy or the Italian people".
The French interior ministry did not respond to a request for comment from CNN.
Following the standoff between Italy and Malta on the migrant rescue on Monday, MaltaToday took to the streets of Valletta to find out what the public thinks on the matter. Conte's office refused to confirm the report.
In televised remarks from Valencia, where Calvo was co-ordinating preparations for the arrival, the deputy prime minister said authorities will examine case by case if the migrants qualify for asylum according to the country's regulations.
A boat loaded with 55 people who were thought to be Syrian refugees landed Wednesday on the northwest coast of Cyprus, a Mediterranean island split into ethnically Greek and Turkish sides.
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.