Pier Luigi Bersani’s Democratic Party (PD) won control of the lower house of parliament in February’s poll, but fell short of a majority in the Senate.
He has ruled out a coalition with Silvio Berlusconi’s centre-right coalition, which finished second.
If no solution is found, Italy could face fresh elections.
President Giorgio Napolitano announced that he had asked Pier Luigi Bersani to form the new government during their meeting at the presidential palace in Rome on Friday.
Giorgio Napolitano said that Pier Luigi Bersani was best positioned to be given the task under “the most difficult circumstances”.
“He will refer back to me as soon as possible,” added the president, who earlier this week held two days of consultations with political leaders.
Pier Luigi Bersani said he accepted the task and would try to “find balance” between changes sought by Italian voters and reforms needed to lead Italy out of the economic crisis.
However, it remains unclear how the centre-left would achieve a majority in the Senate.
Beppe Grillo, whose Five Star Movement won 25% of the vote, has insisted he will not support the PD or any established political party.
Pier Luigi Bersani, 69, is a former Communist who has more recently earned a reputation as an economic reformer.
But he has come under increasing pressure within his own party after failing to convert a poll lead into a clear election victory.
The political uncertainty has increased pressure on Italy’s stagnant economy, as well as feeding into concerns about the stability of the eurozone.
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