The leaders of the two populist parties were furious over the Mattarella veto.
Italy has been without a government since elections in march because no political group can form a majority.
Giuseppe Conte addresses a press conference after meeting with Italian President Sergio Mattarella in Rome, Italy, on May 31, 2018.
That prospect has rocked financial markets, with the euro sinking to multi-month lows on fears that snap elections would lead to a eurosceptic government in Rome.
Italy's premier-designate went to the presidential palace Sunday evening to report on whether he has succeeded in assembling a Cabinet for a populist government that wins the staunchly pro-Europe Italian president's approval while satisfying coalition leaders' demands for a euroskeptic economy minister.
A leaked draft of the so-called "contract" between the two anti-establishment parties indicated that they are willing to leave the prospect of abandoning the euro on the table if the European Union refused to review the fiscal compact.
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The old order in the euro area's third-biggest economy buckled at the March elections as the center-left Democratic Party of ex-premier Matteo Renzi suffered its worst ever result, and Silvio Berlusconi, 81, was eclipsed as leader of the center-right by the more hard-line League.
"Beyond his moral failure to oppose the League's industrial-scale misanthropy, the president has made a major tactical blunder: he fell right into Salvini's trap", Varoufakis wrote in the Guardian.
It wasn't clear what shape a political government might take. In response, government officials expressed solidarity for Mr. Mattarella.
"We are not in the market", Salvini said when asked about the possibility of reopening negotiations, calling for elections as soon as possible. Italy's 2-year government bond yield, which has been the focus of a recent selloff, was down as much as 95 basis points at 1.40 percent.
The head of the anti-establishment 5-Stars, Luigi Di Maio, had proposed a renewed effort to form an alliance with the right-wing League on Tuesday night, backing off his threat of launching impeachment proceedings against Mr Mattarella and insisting that the two blocs still enjoy a parliamentary majority.
Italy is paying a high price for the political chaos of the past few days.
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His cabinet line-up would directly involve M5S leader Luigi Di Maio and League leader Matteo Salvini both in the role of vice prime ministers.
Later on an Italian television, he called for impeaching Mattarella.
Paolo Savona, whose nomination sparked a constitutional crisis, would still join the cabinet, but as European affairs - not finance - minister. The 5-Stars received 32 percent of the vote, making the movement the largest single party in the Italian Parliament.
Italian President Sergio Mattarella is ready to appoint a new prime minister, but he's waiting for a signal from the populists who denounced him as an enemy of democracy.
But a top adviser to Salvini said a breakthrough was unlikely because the League was not prepared to abandon Savona.
But he's also showing some resistance to the change, saying "if someone in Berlin or Paris wakes up in a bad mood that doesn't mean that an Italian minister gets kicked out".
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After rejecting Savona, Mattarella named former International Monetary Fund official Carlo Cottarelli to form a stop-gap government of experts to lead the country to elections.