Mattarella has summoned Carlo Cottarelli, an economist formerly with the International Monetary Fund, for talks Monday, with a temporary technocrat government on the table as Italy faces the strong possibility of new elections in the autumn.
If a coalition can not be formed, the country will again hold elections, with the president having to dissolve parliament and host a vote as early as July 29, if most of the major parties have their way.
In a terse comment to reporters after he relinquished a presidential mandate to put together an acceptable Cabinet, Conte said he "gave the maximum effort, attention, to carry out this task with the full collaboration" of the 5-Star Movement and League.
In another development, far-right Brothers of Italy leader Giorgia Meloni signalled she would support a populist government, reversing her previous position and giving the potential administration more breathing room to pass confidence votes in parliament.
Politics in Italy took another dramatic turn overnight, as the tentative right-wing coalition which formed government in early May unraveled.
M5S leader Luigi Di Maio accused Mattarella's team of lying about Savona's views, and said an impeachment trial for the president was "almost a certainty".
Di Maio has proposed moving contested ministerial candidate Paolo Savona to a different Cabinet post.
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That is what ultimately happened after Mattarella on Sunday night refused to approve the populists' choice of an economy minister who has raised the spectre of Italy having to someday exit from the euro, the European Union currency shared by 19 countries.
Cottarelli, 64, was director of the IMF's fiscal affairs department from 2008 to 2013 and became known as "Mr Scissors" for making cuts to public spending in Italy.
Matteo Salvini, the leader of the League, said the appointment of Mr Cottarelli was the death throe of a political establishment intent on keeping Italy "enslaved".
The IPSOS poll, in the Corriere della Sera newspaper, showed support for the League's would-be coalition partner, the 5-Star Movement, steady at about 32.6 percent - implying a much more comfortable majority if the pair were to try again to govern.
Mattarella welcomed the proposal, according to his staff. League leader Matteo Salvini, who is surging in opinion polls, seemed cool on the idea but did not rule it out. The upshot is that while Cottarelli's technocratic government team is ready, he and Mattarella are allowing more time for the populist parties to seek an agreement, the official said.
"In a democracy, if we are still in democracy, there's only one thing to do, let the Italians have their say", Salvini said in a fiery speech to supporters in central Italy.
However, Mr Cottarelli admitted his role might not even last that long if he does not win confidence votes in both of Italy's parliamentary chambers.
"The next elections will be a plebiscite, the people and real life against the old castes and the Lords of the Spread", Salvini said, referring to financial speculators, in a statement issued by his League.