Macron said that "Italy has had massive migrant arrivals, but in the previous year we have reduced them by 77%".
The French and Italian leaders have agreed to work together to push for changes to the European Union's migration rules after previously disagreeing on the issue. At least 12 other people died when their dinghy capsized.
Italy then triggered a row with European nations when it refused to allow the Aquarius, a rescue ship with 629 migrants aboard, to dock at its ports.
France also said it was willing to take in migrants from the Aquarius after they arrive in Spain if they "meet the criteria for asylum".
Calls and emails to the 6th Fleet were not answered late Friday.
The French president, Emmanuel Macron, in an attempt to retain Italy in club of most pro-Europeans, has done this Friday common front in immigration crisis with Italian Prime Minister Giuseppe Conte.
They have either completely opposed or resisted quotas that European Union member-countries endorsed in 2015, at the peak of the migration crisis, when more than 1.26mn people applied for asylum in the bloc.
"Ships belonging to foreign organisations and flying foreign flags can not dictate Italy's immigration policy", Salvini, who also heads the anti-immigrant League party, said on Wednesday.
"The proper response is European, but the existing European response has not adapted", said the French leader, who is seen as strongly pro-EU. The concept itself of the state of first entry will be rethought.
On Wednesday, Italy had summoned France's envoy and demanded an apology from Macron, who had said Rome's move to block the ship from its ports was an act of "cynicism and irresponsibility".
They both agree that the Dublin Regulation should change.
"France past year received a record number of asylum seekers", she said, with 100,000 requests.
French President Emmanuel Macron sought to resolve tensions with Italy on Thursday, saying he never meant to cause offence by criticising its rejection of a migrant ship in remarks that sparked fury in Rome.
Discussions were complicated by a report released Friday by Oxfam, highlighting the plight of almost 20,000 migrants languishing at the Italy-France border without adequate water, food, shelter, or counsel on their rights.
The exchange of barbs took place a day after Italy and France tried to bury the hatchet following a diplomatic squabble over Rome's refusal to accept the Aquarius.
Prime Minister Pedro Sanchez of Spain, where the Aquarius is expected to arrive on Sunday, called on Twitter for "a new Europe where national egoism does not prevail".
"It's very clear that Europe needs to reform its asylum system in a collective manner; there's a lot of resistance to that, but there's no other way", Grandi said.
In Germany, Chancellor Angela Merkel is embroiled in a showdown with the right wing of her governing coalition, which is demanding that she immediately ditch her liberal migration policy and tighten border controls.
Pope Francis referred to this week's standoff for the second time in as many days Friday, saying the Gospel teaches that it's wrong to leave migrants "at the mercy of the waves".