Macron, who spoke by telephone to Italian President Sergio Mattarella on Tuesday to “reaffirm the importance” of bilateral ties, asked the French ambassador to deliver the invitation at a meeting Friday evening, the Elysee Palace said.
Mattarella, a centre-left politician, is an elder statesman whose job as president carries limited political powers.
France had announced on February 7 that it was recalling its ambassador, Christian Masset, to protest “unfounded attacks and outlandish claims” by Italy's populist coalition government — led by deputy prime ministers Luigi Di Maio and Matteo Salvini.
“I am very happy that the ambassador is on his way back to Italy,” Luigi Di Maio, who is deputy prime minister, told reporters in Rome. “I shall meet him, I want to ask him for a meeting.”
Relations between the two countries have fractured due to repeated clashes between Di Maio and Salvini's populist coalition government and France's centrist Macron.
Paris was incensed when Di Maio made a surprise visit to France on February 5 to meet a group of radical “yellow vest” protesters who have led demonstrations against Macron.
“The wind of change has crossed the Alps,” Di Maio wrote afterwards of the three months of protests against Macron, adding that he was preparing a common front ahead of European Parliament elections in May.
French Foreign Minister Jean-Yves Le Drian said a “line was crossed” with the visit, which was organised without French authorities being informed. The last time Paris recalled its ambassador to Rome was during the World War II when Italy under leader Benito Mussolini invaded France in 1940.
The current icy ties between two founding members of the European Union has many analysts wondering about the consequences for the bloc, given that French-Italian ties have been a generally stable axis.
It risks complicating a major infrastructure project between the countries that would result in a tunnel being bored under the Alps to link the important regional cities of Lyon and Turin.
Work on the 57.5-kilometre (36-mile) tunnel, set to cost an estimated 8.6 billion euros (9.7 billion dollars), is suspended pending a green light from the Italian government.
Di Maio's party, the Five Star Movement, is opposed to the project, while its coalition partner the far-right League party, headed by Salvini, who is also interior minister, is in favour.
“France clearly respects the time that our Italian partners wanted to take. But today we are saying clearly to the Italians that this decision needs to come,” French Transport Minister Elizabeth Borne told the Public Senat channel on Friday.
Analysts and diplomats say that relations have been affected by the fundamentally different outlooks of Macron, a pro-European centrist, and the eurosceptic government in Rome.
There are also deep-running economic tensions, competition for influence in Libya, and a sense in Italy that France has done little to help its neighbour cope with the arrival of hundreds of thousands of migrants in recent years.
Posturing ahead of the elections for the European parliament have exacerbated these tensions, observers say.
A French diplomat, speaking on condition of anonymity, said that Di Maio and Salvini's recent criticism of Macron and France was driven by competition between the two men.
“Di Maio and Salvini are in competition against each other. Their vision is that at some point there will be only one of them,” the diplomat said, adding that the European elections in May would be vital.