Mattarella had given Conte a mandate to form a new government last Wednesday, after he was put forward by the euroskeptic rightwing League and the populist Five Star Movement, the two top vote-getters in Italy's inconclusive March 4 election, as their choice for prime minister.
Italy, the eurozone's fourth-biggest economy, has been without a government since elections in March because no political group can form a majority.
President Sergio Mattarella has summoned former International Monetary Fund director Carlo Cottarelli for talks at 11:30, where they'll discuss the possibility of a temporary technocratic government.
This was the first time in the history of the Italian republic that a president had thrown out the proposal for a government from elected parties, and even critics of the populist victors have acknowledged that the Italian people will take the move as a slap to Italian voters and the entire democratic process.
Giuseppe Conte, a little-known law professor, had been nominated by the anti-immigrant League and the anti-establishment Five Star Movement to head their coalition government.
The euro crawled off a 6.5-month low against the dollar on Monday, catching its breath after Italy's president tried to allay investor worries about political unrest in the country, although the prospect of a near-term election capped gains.
Mattarella said he has done "everything possible" to aid the formation of a government, but that an openly eurosceptic economy minister ran against the parties' joint promise to simply "change Europe for the better from an Italian point of view".
Data last week showed German PMI data fell to a 20-month low in May indicating that economic momentum in Europe's biggest economy was faltering and European Central Bank minutes of its April meeting showed policymakers were anxious about a more pronounced slowdown in the eurozone and political uncertainty in Italy.
The country could be headed for another election in coming months.
Mattarella said he had accepted every proposed minister except Savona, who has called the euro a "German cage" and said that Italy needs a plan to leave the single currency "if necessary".
Both Salvini and Five Star leader Luigi Di Maio reacted furiously to the president's intervention on Sunday evening, with Di Maio blaming credit-rating companies for torpedoing the cabinet.
The president explained in a televised address his decision to block the appointment as finance minister of 81-year-old Eurosceptic economist Paolo Savona.
Cottarelli would be a calming choice for the financial markets, but any technocratic administration would likely only be a short-term solution because the majority of parliamentarians have said they would not support such a government.
The parliamentary election, held on March 4, failed to produce a clear victor but saw support for traditional parties collapse. "It seems a stretch", Mr Salvini said.