President of the Italian Senate, Maria Elisabetta Alberti Casellati (L), and President of the Italian Chamber of Deputies, Roberto Fico (C) count ballots in the Italian parliament in Rome on January 25, 2022. AFP
More than half of the almost 1,000 MPs, senators, and regional representatives who voted left their papers blank for a second day, reflecting the lack of agreement on a candidate among the main parties.
A third round of voting will now be held Wednesday morning, although no breakthrough is expected until Thursday.
From the fourth round onwards, the threshold for victory falls from a two-thirds majority to an absolute majority.
Italy's presidency is a largely ceremonial role but the contest this year has high stakes, as Draghi is tipped for the job.
His move would unsettle the fragile coalition, risk snap elections, and potentially derail reforms required for billions of euros in EU recovery funds.
However, the presidential vote is notoriously hard to predict, with secret ballots, backroom deals, and lack of a formal candidate list drawing comparisons with a papal conclave.
No political grouping has a majority in parliament. Instead, almost all the parties, from left to right, share power in a national unity government.
Draghi, a former European Central Bank chief, was brought in by outgoing President Sergio Mattarella in February 2021 as Italy reeled from a pandemic-induced recession.
His government has overseen a return to growth and a successful coronavirus vaccination campaign.
And he has begun major reforms -- notably to the tax and justice systems and public administration -- demanded by Brussels in return for almost 200 billion euros ($224 billion) in EU grants and loans.
Many international investors are concerned that debt-laden Italy would slip behind on the tight reform schedule should Draghi leave.
There are also many Italian MPs who fear losing their seats if his exit sparks early elections.
Others say Draghi would be better placed as president to ensure political stability and good relations with Brussels -- particularly should the far-right win the next election.
The head of state wields considerable power during political crises, from dissolving parliament to picking new prime ministers and denying mandates to fragile coalitions.
Paolo Maddalena, a little-known former judge who led the field in Monday's voting, topped the list again on Tuesday with 39 votes.
An equal number voted for Mattarella, 80, despite his making clear he does not intend to serve a second seven-year term.