Letta said he accepted the mandate with "reservation," noting Italy was in a "new, fragile situation that cannot continue," the Italian news agency ANSA reported.
"I accepted feeling a great responsibility on my shoulders," Letta said. "The country is waiting a government."
Napolitano, re-elected president Saturday, said he would move quickly to name someone who could run a coalition government to try to end a two-month political impasse and set the country on course for reforms.
The meeting between Napolitano and Letta was seen as the first step to Letta being named prime minister, The Wall Street Journal reported.
Napolitano said his decision to give Letta the mandate "opened the door" to the "government that the country urgently needs."
The president said the only governance possibility was "a broad agreement between the political parties that can guarantee a majority."
Napolitano called on Italy's political parties to cooperate "to create a climate of maximum detente rather [than] the old tensions."
Former Prime Minister Silvio Berlusconi's People of Freedom Party warned the Democratic Party it had to support the coalition amid concerns internal rifts within the Democratic Party would result in a weakened government, ANSA said.
"If it's a question of having any little government that risks being short-lived, they can do what they want, but we won't be part of it," People of Freedom Party Secretary Angelino Alfano said of the Democratic Party. "We have the clear impression that the [Democratic Party] doesn't want to create a strong government."