Mr Macron's office said France does not want to "start a precedent" and allow European countries to breach global laws and rely on other member states to take in migrants, who must now apply for asylum where they first land in Europe.
Italy's new economy minister Giovanni Tria had been due to meet his French counterpart Bruno Le Maire in Paris Wednesday afternoon.
A rescue ship with 629 migrants on board that is on its way to Spain has become a "symbol of the failure of the European Union", the head of the Spanish branch of Doctors without Borders (MSF) which helps operate the vessel said on Thursday. Foreign Ministry spokeswoman Agnes von der Muhll said France was "fully aware of the burden that the migration pressure is placing on Italy" and that it was committed to cooperating with Rome on immigration.
The Aquarius was on its way to Valencia with an escort of two Italian ships after the Italian government denied it access to its ports at the weekend in a case that has caused political tension in Europe.
The Spanish government said it is expecting ships carrying 630 migrants to arrive in the port of Valencia on Sunday morning instead of Saturday after bad weather forced the convoy to take a detour.
Valencia is nearly three days' voyage for the Aquarius, while Italy and Malta are just hours away. Malta had refused permission for the Aquarius to berth in Malta, insisting that it was not the competent authority. SOS Mediterranee said they had received food aid and supplies from the Maltese Navy and Italian Coast Guard.
Migrant-related issues will be at the heart of the discussions and France hopes to maintain a "close dialogue" with its neighbor, the statement says.
Salvini said in three years, France has accepted only 640 people. We can not guarantee that it is suitable for the visually or hearing impaired.
Matteo Salvini accused France of turning back 10,000 migrants on its border with Italy.
The country has seen more than 700,000 migrants arrive on its shores since 2013.
Grandi said closing ports wrongly threatens "rescue at sea" - a requirement under global law - but said Italy's reasoning was "something that we need to listen to".
An Italian coastguard ship carrying more than 900 migrants was allowed to dock in Sicily on Wednesday.
Aquarius had conducted the rescue operation off the coast of Libya on June 9.
The Trenton initially had asked a private rescue ship off Libya, the Sea Watch-3, if it would take aboard the survivors.
USA officials didn't respond to requests for information about where the migrants would go, saying only that they were being provided with food, water and medical care.
Mr Macron attempted to defuse the row with the Italian government on Wednesday, saying in a speech in the western French town of Mouchamps that France had been "working hand in hand with Italy" to better manage migration flows for more than a year.
Conte said the chat with Macron had been "very cordial" and confirmed Friday's meeting in Paris.
Neither office gave a reason, but earlier the Italian Foreign Ministry said it had summoned the French ambassador to complain that the French comments about the Aquarius were "unacceptable" and were compromising their diplomatic relations.
It added that Italy was waiting for French to take action to "heal the situation that has been created".
A police official, who spoke on condition of anonymity because he was not authorized to speak publicly, told the Associated Press there were 54 men and one woman aboard the boat.