According to tradition, the president will not ask her to form a government before holding formal talks with all parties in parliament.
She could be sworn in this weekend, with a confidence vote in parliament next week.
If Meloni succeeds, she will become Italy's first female prime minister.
The Brothers of Post-Fascist Italy won 26 percent of the vote in Italy's parliamentary elections, becoming the largest party.
It has thus taken the lead in its alliance with Berlusconi's party Forza Italia and the other major far-right party Lega.