Conte arrived at the Elysee Palace in Paris for afternoon talks ahead of a press conference, with both smiling as Macron greeted the Italian leader.
Despite efforts by both sides to play down testy exchanges in recent days, the clash underscores deep divisions in Europe over how to handle the massive influx of migrants from across the Mediterranean in recent years.
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.
An opinion poll by Infratest Dimap suggests 62 percent of Germans back the tougher stance favoured by her interior minister, including turning back undocumented migrants at the border and deporting rejects faster.
Merkel has pleaded for more time to negotiate with her European partners on a common response ahead of an EU summit on June 28-29.
But there are few signs that European leaders are anywhere near being ready to coordinate their policies despite a looming end-of-June deadline to change the EU's current asylum rules.
Italy 'pushed around'
The Aquarius rescue vessel at the centre of this week's row was continuing on Friday to make its way across the Mediterranean to Spain, which agreed to take the 629 migrants aboard after Italy and Malta refused the ship permission to dock.
Italy's hardline new interior minister Matteo Salvini said he hoped the boat would get “a rousing welcome” in Spain, adding that his country was set to present ideas for migration reform to European partners.
“I have already shared a fundamental idea with the Germans and the Austrians — that of defending external borders with men and money, including the Mediterranean,” he said.
“We have had enough of being pushed around by others,” he told residents in the northern Italian city of Genoa.
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”.
Under the EU's Dublin Agreement, which is currently up for review, migrants hoping to apply for asylum must do so in the first country they enter, a policy which has placed a huge burden on Italy in particular.
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.
“We need to work on reform of the Dublin Agreement,” Conte stressed ahead of his Paris visit.
Earlier this week Salvini joined forces with the German and Austrian foreign ministers in an “axis of the willing” to combat illegal immigration.
Other countries meanwhile, such as Hungary, Poland, the Czech Republic and Slovakia, have either refused outright or resisted taking in refugees under a contested EU quota system.
The spat between France and Italy erupted this week after Macron accused Rome of “cynicism and irresponsibility” for refusing to let the Aquarius dock.
Italy's new government hit back, accusing Paris of giving “hypocritical lessons” and threatening to pull out of the meeting with Macron.
Rome also summoned the French ambassador over the dispute — the second time it has done so over the migrant crisis in two months.
Macron's critics said he was hardly in a position to lecture, noting that France had taken in far fewer migrants since the start of the crisis than the likes of Germany and Sweden, and has sealed off its border to most migrants trying to cross into the country from Italy.
The French leader, who has taken a tough line on migration from African countries that are not at war, said Thursday that “none of his comments were intended to offend Italy”.
In a further gesture of reconciliation the French foreign ministry said it was ready to welcome migrants aboard Aquarius who “meet the criteria for asylum” after they arrive in Spain.
Italy itself has appeared eager to avoid too harsh a response. After turning the Aquarius away, it allowed a coast guard ship carrying over 900 migrants land on Sicily on Wednesday.
And on Thursday, the Italian coast guard brought fresh supplies to the Aquarius as it made its way past Sardinia.
By Laurence Benhamou and Clare Byrne