The political novice's task was complicated by right-wing leader Matteo Salvini's insistence that the Cabinet include his pick for economy minister, a former government minister who has raised doubts about Italy keeping the euro as its official currency.
Shortly afterwards, the president's office summoned Mr Cottarelli, the IMF's former director of fiscal affairs, for a meeting on Monday.
Mattarella became the target of populist rage when he said he rejected the populists' choice of Paolo Savona for finance minister for the good of the country and the financial "savings of families" that had been endangered by rising bond spreads and market concerns.
Salvini, along with 5-Star Movement leader and fellow euroskeptic Luigi Di Maio, agreed this month to join their rival forces in a coalition to break a political impasse caused by an inconclusive March 4 parliamentary election.
The former International Monetary Fund official said that he would plan for general elections after August if his government loses parliamentary votes of confidence, or in early 2019 if it wins the ballots in the lower house and in the Senate.
"Now I am telling Mattarella that we want a date for new elections".
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. He added that the League and 5-Star had refused to put forward any other name for the role. "Tomorrow we want to know the date of the election", Salvini said on Sunday during a live stream on his Facebook page.More news: Nadal powers into Italian Open third round
The greenback was up 0.05 per cent at 109.400 yen after going as high as 109.830 on a slight ebb in risk aversion after U.S. President Donald Trump said on Sunday a U.S. team had arrived in North Korea to prepare for a proposed summit between him and North Korean leader Kim Jong Un.
"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.
For anyone wondering how the Italian president, usually seen as acting as a figurehead, was able to veto a minister backed by most parliamentarians, here's an explainer of how much power the president has in these situations.
Such a solution could only be short term, as most members of parliament have said they would not support such a government. Italian heads of state have made bold moves in the past. However, previous presidents have also challenged individual ministers.
The Five Star's Di Maio demanded impeachment under article 90 of the constitution.
In his first remarks, Cottarelli sought to calm markets and Italy's European partners, assuring that his government would guarantee "prudent" management of Italy's debt, which at 132 percent of gross domestic product is Europe's heaviest after Greece.
"The designation of the finance minister always sends an immediate message, of confidence or of alarm, for economic and financial operators", he added.