An angry Di Maio, of the 5-Star Movement, made the unprecedented demand after Mattarella rejected a euroskeptic for the key post of economy minister - virtually foiling a bid to form Europe's first fully populist government.
Mattarella's objection to the economy minister candidacy earlier drew the ire of MS5 leader Luigi Di Maio, who called the president's decision "incomprehensible" and called for his impeachment.
This prompted Italian anti-establishment parties 5-Star Movement and League to abandon plans to form a government.
Mr Mattarella, however, objected to the choice of Paolo Savona, 81, as economy minister, forcing the prime minister-designate to abandon his efforts to form a government, more than 80 days after the elections.
The constitution also gives the president the power to dissolve parliament, a deterrent force which has played a part in numerous prior political crises in Italy - a country which has had 64 governments since 1946.
The president said he approved all of the other Cabinet picks, but rejected the coalition partners' choice for the economy portfolio out of concern it would have a negative effect on financial markets and the Italian economy.
Newspaper La Stampa reported earlier that Conte, a law professor at Florence University with no political experience, will try to persuade and reassure Mattarella that Savona won't seek a euro-exit or create problems with Brussels.
He had watched for weeks as Five Star and the League set about trying to strike an alliance that would give Italy's hung parliament a majority. In a new book, Savona, described Italy's adoption of the euro as a "historic error", according to media accounts.More news: Hawaii's Kilauea volcano spews lava
Mattarella's move is being widely interpreted as a rejection not only of the government but of the national will as expressed in democratic elections, and massive demonstrations are expected Monday.
Cottarelli, known as "Mr Scissors", has been summoned for talks with the president later Monday.
"The uncertainty over our position has alarmed investors and savers both in Italy and overseas", Mattarella said, adding: "Membership of the euro is a fundamental choice".
"If there's not the OK of Berlin, Paris or Brussels, in Italy a government can not be formed".
"Either the government gets off the ground and starts working in the coming hours, or we might as well go back to elections", Salvini said.
An administration led by Cottarelli would likely only be a short-term solution because the majority of parliamentarians have said they would not support a technocrat government.
A former judge of Italy's constitutional court, Mattarella has refused to bow to what he saw as "diktats" from the two parties which he considered contrary to the country's interests.
Last week, the spread of points between Italy's bonds and benchmark German bonds grew alarmingly, and Milan's stock market suffered losses as investors were spooked about the intentions of the populists.