Mr Prodi, a 66-year-old former economics professor, said he had accepted the request "with reserve," explaining that he had yet to reach agreement within his disparate nine-party coalition on his ministerial line-up.
He said he hoped to be able to finalise a government line-up early on Wednesday.
"I guaranteed the president that we will work to form a balanced and solid government able to face the political and economic problems of the country," he said.
Italian media reports indicated smaller parties in Mr Prodi's Union coalition were digging in their heels over who should get the defence and interior ministry portfolios.
President Napolitano said earlier that he had decided to ask Mr Prodi to form a government after a day of consultations with party leaders and former presidents -- a formality required by the constitution.
But out-going prime minister Silvio Berlusconi promised the new government a tough time from his coalition, which now goes into opposition.
"We will give no help to the centre-left government, an executive which will not have an easy time, and which in the Senate will be able to win a confidence vote only through the crutches of the senators for life," said Mr Berlusconi in his final press conference as prime minister.
Mr Berlusconi, ousted last month after leading Italy's longest-serving, post-war government, said his coalition "will be ready to profit, democratically, from the weakness" of Mr Prodi's government.
The new government will face a vote of confidence in the Senate -- where it holds a tiny majority -- before the end of the week, and a similar vote in the lower house of parliament early next week.
A defiant Mr Berlusconi continued to contest the election result, insisting that "many irregularities" still needed to be cleared up.
"I nourish a lot of hope on a recount of the ballots and I'm convinced that the result of the polls can be overturned," he said. A parliamentary commission on the elections has yet to examine the disputed polls.
The billionaire media magnate said his political forces would "use all foreseen democratic methods" to have the result overturned, in which case he said the president would have to accede to the opposition's request for snap elections.
Mr Prodi's Union coalition won the closest election in living memory by less than 25,000 votes in the lower house, and holds a majority in the upper house of just two seats.
The former economics professor indicated earlier that he had surmounted disputes within his coalition over who should get prominent ministerial and deputy prime ministerial posts.
"The team is ready," he told journalists after more than an hour of talks with the head of state during the cross-party consultations.
"I reassured the president of the republic that our coalition is solid," Mr Prodi said.