Napolitano is expected to give a government-formation mandate to the leader of Letta's Democratic Party, Florence Mayor Matteo Renzi, the Italian news agency ANSA reported.
Napolitano said he was consulting with parties on forming a new government "in the shortest time possible" after Letta quit.
If he secures the support of Parliament, the 39-year-old Renzi would be Italy's youngest prime minister.
On Thursday, the Democratic Party backed Renzi to try to form a new government after he said it was time to pull Italy out of a "morass" of stalled reforms and economic woes, ANSA said.
"This is one of the best moments in my life," Renzi told reporters in Florence.
Renzi thanked Letta for his "remarkable" work maintaining fiscal order but said Letta didn't deliver on promises of a new electoral law and stronger stimulus package for Italy's slowly recovering economy.
Letta announced his resignation Thursday after a 10-month tenure marked by frustrated economic reform efforts and struggles to hold together two left-right coalitions.