The Spanish authorities then said they were ready to let the Aquarius dock in one of their eastern ports.
But the tensions appeared to have ebbed after a telephone call Wednesday night between Macron and Conte.
French Foreign Minister Jean-Yves Le Drian spoke with his Spanish counterpart Josep Borrell over the phone about the current European migrant crisis, and both sides urged "a broad discussion" of the hard issue at the European Union summit on June 28-29, the French Foreign Ministry said in a statement.
French President Emmanuel Macron denounced the block on the Aquarius, insisting that Italy should have taken the migrants in.
A European Union summit later in June is due to consider changes to asylum law to better share the burden of incoming migrants.
Conte is "tending toward pushing back the trip to Paris", the source said.
Their talks are meant to focus on the contentious issue of migration.
SOS Mediterranee said men who were still on the deck of the Aquarius exposed to the elements were dizzy and vomiting and were taken inside the ship to be treated for seasickness.
"The EU's own survival depends on the handling of migration", Antonio Tajani tweeted, calling it "the biggest challenge of our times".
The tweet confirmed that Tria's meeting with German Economy Minister Olaf Scholz scheduled for Thursday would go ahead in Berlin.
Fed hikes interest rates again, raises outlook for more increases in 2018
Powell has repeatedly played down the dot plot as a guide to future interest rates, though investors continue to focus on it. Investors had given just over a 91 per cent chance of a rate rise on Wednesday, according to an analysis by CME Group.
The ministry issued a statement after Foreign Minister Enzo Moavero Milanesi summoned the French ambassador to complain on Wednesday.
"I'm glad Spain has stepped forward to defuse this crisis, but I fear a major tragedy if states start refusing to accept rescued migrants as was threatened", said IOM's Director General William Lacy Swing.
Salvini has repeatedly vowed to stem the flow of migrants to Italy, which has seen more than 700,000 arrive on its shores since 2013.
Most of the over 600 migrants who are on board the Aquarius have been transferred to the escort vessels.
New Italian deputy Prime Minister Luigi Di Maio hit back angrily saying "I'm glad the French have discovered responsibility, if they want, we will help them".
Under EU rules, migrants must apply for asylum in the European country where they first arrive.
France tried to take a more conciliatory tone on Wednesday.
Salvini has accused European aid groups of essentially operating taxi services for Libya-based human traffickers, and he has said Italy will now refuse rescue ships' entry.
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.
In a swipe at Macron, Italy's government said Tuesday it would not accept "hypocritical lessons from countries that have preferred to look the other way on immigration".