Mattarella, 80, had ruled out remaining in office but with the country's political stability at risk, it was considered unlikely he would resist pressure to stay on.
Mattarella's "willingness to serve a second term... shows his sense of responsibility and his attachment to the country and its institutions," Regional Affairs Minister Mariastella Gelmini said in a statement.
The leader of the Democratic Party (PD) Enrico Letta spoke to reporters to express his "enormous thanks to President Mattarella for his generous choice towards the country".
Parliamentary chiefs went to the president's palace in central Rome to ask him to remain in office, after lawmakers failed to elect a president on Saturday after seven rounds of voting.
There was no immediate comment from the president himself.
Prime Minister Mario Draghi, who failed to find backing for his own ambitions for the job, earlier called Mattarella and also urged him to stay on, a political source said.
The move followed five days of stalemate in parliament, with the main political blocs incapable of agreeing on a candidate who could win broad backing among lawmakers.
"The Italians do not deserve any more days of confusion," Matteo Salvini, head of the League party, said.
"Let's reconfirm President Mattarella... and Draghi, and immediately get back to work this afternoon. The problems of the Italians won't wait," Salvini added.
It is the second time in succession that a president has been asked to renew his seven-year mandate.
In 2013, political leaders appealed to the then head of state, Giorgio Napolitano, who was almost 90, after they too failed to find a consensus candidate.
Napolitano reluctantly agreed but stood aside two years later after a new government was installed.Â
Mattarella will almost certainly do the same once the political situation allows it, many commentators have said.
The fruitless efforts to replace him have left deep scars on the parties and their leaders, with the centre-right alliance in particular disarray after losing any semblance of unity over the past 24 hours.
This could have repercussions for the stability of Draghi's broad coalition government, which includes parties of left and right that have been at loggerheads over the election of a new president.
"The political landscape has changed," said Letta, who has emerged stronger from the week of political manoeuvring and may seek more weight in the cabinet for the PD.
While both Salvini's League and Forza Italia in the end embraced the prospect of maintaining the status quo, their ally the Brothers of Italy, which has not joined them in the ruling coalition, denounced the behind-the-scenes manoeuvring.
"Parliament has shown it is not fit for Italians," said Brothers of Italy leader Giorgia Meloni in a statement, accusing her allies of "bartering away" the presidency to ensure the government remains in place until the legislature ends in 2023.
The president is a powerful figure who gets to appoint prime ministers and is often called on to resolve political crises in the euro zone's third-largest economy, where governments survive around a year on average.
Unlike in the United States or France, where heads of state get elected in a popular vote, in Italy 1009 parliamentarians and regional representatives chose via a secret ballot that party leaders sometimes struggle to control.
Threatening to take charge of the situation themselves, lawmakers have been increasingly backing Mattarella in the daily ballots, with his tally rising to 387 on Saturday.
A successful candidate needs 505 votes to win and Mattarella looked set to comfortably pass the threshold at the eighth ballot.
