The meeting follows a surge in risk spreads in Italian 10-year bonds to more than 300 basis points this week after the coalition submitted a defiant budget plan.
This sent tremors through the Italian banking system - highly sensitive to bond rates - and risks setting off a credit crunch.
The fiscal revisions released last night are slightly less radical but they still breach the Eurozone's stability pact.
Luigi di Maio, the Five Star leader and Italy's vice-premier, said: "We're watching the markets - but if there is a choice between the bond spreads and citizens, I will choose the citizens."
La Stampa said Draghi expressed alarm over the budget fight, warning that politicians in Rome are vastly underestimating the delicacy of the situation as the ECB winds down its bond purchase programme over the next three months.
"Italy will have no safety net," he is reported to have said.
There is a backstop rescue facility known as the OMT but this requires involvement by the EU bailout machinery (ESM), under draconian conditions, with a vote in the German Bundestag.
This option would not come in to play until the crisis was already far advanced, and even then it is doubtful whether the Lega-Five Star alliance would ever accept such terms.
Mattarella, a pro-EU centrist of the old guard, has some say over budget compliance and could theoretically invoke his constitutional powers to uphold Italy's membership of the euro.
Officials are studying clauses in the constitution to determine the exact extent of his prerogatives.
If the independent budget office (UPB) issues a harsh verdict on the fiscal forecasts over coming days, or warns of any threat to the country's long-term debt sustainability, this could strengthen the president's hand.
"This is complete madness," said Borghi.
"They can't do it unless they send in the army and you can't do this to a lawful government with a majority in parliament that is doing nothing out of the normal. If they try, there will be a real revolution."
The coalition is infuriated by a Reuters story citing anonymous EU officials and diplomats in Brussels warning that Italy faces "massive debt restructuring" on its €2.3 trillion ($4.1t) public liabilities on the current course.
One described the Italian budget as "complete insanity", saying it relies on "ludicrous growth assumptions".
Another said delusional markets were clutching at straws and had not yet grasped the full gravity of the clash between Rome and Brussels.
As always in such cases it is hard to tell how much of this is a pressure tactic; an attempt by the European Commission to recruit the bond vigilantes to scare Italy and act as their enforcement gendarmes.
"If there is a deliberate political and financial attack on our country, we will defend ourselves," said Borghi.
"The Italian people went through this before in 2011 and they are no longer afraid of the spreads."
He declined to comment on the government's "plan B" defences, but earlier proposals for a "minibot" parallel currency are written into the official coalition contract - should it be needed to inject liquidity into the banking system and prevent the sort of asphyxiation that brought Greece to its knees in 2015.
"All I will say is that the EU should have learnt after what happened in Greece that once you start talking about debt restructuring, you are igniting the fire," he said.
The budget dispute between Italy and the EU is in one sense contrived, more political theatre than economic science.
The latest plans project a deficit of 2.4 per cent of GDP in 2019, 2.1 per cent in 2020 and 1.8 per cent in 2021. This is hardly a shocking figure.
France is still near 3 per cent and has been a persistent violator but is forgiven "because it is France" in the unforgettable words of Commission chief Jean-Claude Juncker.
In part, the showdown is over EU theology. "The rehabilitation of the Eurozone's fiscal rules is at stake," said Neil Mellor from BNY Mellon.
Juncker said that if Italy was granted a free pass on the revamped (and contractionary) stability pact, it "would mean the end of the euro, so you have to be very strict".
At the same time there is a strong suspicion in Rome that the EU is raising the stakes to teach Italy's Eurosceptics a lesson.
Di Maio has come close to accusing France and Germany of conspiring to bring about regime change.
Yet it is also clear that Italy's budget figures are fiction. They rely on a surge in economic growth of 1.5 per cent to lower future deficits.
Daniele Antonucci, from Morgan Stanley, said the best they could hope for was 1 per cent growth.
Brian Coulton, from Fitch, said the trend rate was no higher than 0.7 per cent.
Reforms are going backwards so, if anything, the speed limit may fall further.
The Lega-Five Star government argues that years of depression have left Italy with a large "output gap", meaning any fiscal stimulus will turbo-charge the economy through a high fiscal multiplier.
But this is contested. The "hysteresis" damage from the slump has allowed labour skills to atrophy and may have destroyed part of the economy permanently.
Both the Commission and the OECD say the output gap has already been closed. They argue that there is no slack, and no such low-hanging fruit to pick.
Lorenzo Codogno, former chief economist at the Italian Treasury and now at LC Macro Advisors, said Italy was still "miles from compliance" under the revised budget figures.
Moody's and Standard & Poor's will issue their verdict on Italy's sovereign debt rating over the next month. A one-notch downgrade might be manageable.
If the markets start to fear further slippage to junk status, it would set off a mass exodus by foreign investors.
The Norwegian pension fund, the world's biggest investor, said yesterday that it would be forced to liquidate its holdings of Italian debt under its mandatory guidelines if the country falls below investment grade.
The lesson of the Greek crisis is that it does not pay to be the last out of the door when things go wrong.
