Billionaire former premier Silvio Berlusconi withdrew from the contest on Saturday, but despite continued wrangling over the weekend, no clear candidate has yet emerged.
Draghi is facing opposition from Berlusconi's Forza Italia party and also Matteo Salvini of the anti-immigration League party, who says he should stay where he is.
"It would be dangerous for Italy in a difficult economic time... to reinvent a new government from scratch. It would stop the country for days and days," Salvini told reporters on Sunday.
But Enrico Letta, leader of the centre-left Democratic Party, said Draghi had been an "extraordinary resource" for Italy and insisted talks would continue, telling Rai television: "Draghi is one of the hypotheses on the table."
Italy's presidency is largely ceremonial, but the head of state wields considerable power during political crises, from dissolving parliament to picking new prime ministers and denying mandates to fragile coalitions.
The election, a secret ballot conducted over several days by more than 1,000 MPs, senators and regional representatives, is notoriously hard to predict.
Draghi, a former European Central Bank chief brought in to lead a national unity government one year ago, is widely considered the most eligible candidate.
But many fear his departure as premier could trigger chaos as Italy recovers from the devastation of the coronavirus pandemic.
With the disparate parties in Draghi's coalition already in battle mode ahead of next year's general elections, further instability could put European recovery funds at risk.
"This is a key and very complicated election, because the political parties are weak, they are in an utterly fragmented state," Giovanni Orsina, head of the Luiss School of Government in Rome, told AFP.
Italy has a notoriously unstable electoral system and has seen dozens of governments come and go since World War II -- with outgoing president Sergio Mattarella himself seeing five during his seven-year term.
But Draghi, brought in by Mattarella in February 2021, has led a remarkably united government comprising almost all of Italy's political parties.
Italy, the eurozone's third largest economy, has returned to growth following a punishing recession in 2020 sparked by the pandemic.
And Draghi has initiated key reforms demanded in exchange for funds from the EU's post-pandemic recovery scheme, of which Rome is the main beneficiary, to the tune of almost 200 billion euros ($225 billion).
Many international investors are concerned that debt-laden Italy would slip behind on the tight reform schedule should Draghi step down as prime minister.
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 74-year-old himself, credited with saving the euro from a debt crisis while ECB chief, hinted last month he is interested in moving to Rome's Quirinale presidential palace but has since kept his silence.
Car park vote
The first round of voting begins at 3:00 pm (1400 GMT) on Monday in the lower Chamber of Deputies, with its result expected in the evening.
Voting is in secret, in person and will be slowed by social distancing requirements, with one round a day.
Commentators predict no breakthrough until Thursday, the fourth round, when the threshold for victory falls from a two-thirds majority to an absolute majority.
Because of Italy's high Covid caseload, electors who tested positive or are isolating will be able to use a drive-through voting station set up in the parliament's car park.
Berlusconi's presidential bid was always a long shot, not least because he remains embroiled in legal proceedings over his "Bunga Bunga" sex parties.
He said he was quitting out of a sense of "national responsibility", but media reports suggested his family were worried about his health.
He has been in and out of hospital since a serious bout of Covid in September 2020 and was admitted to Milan's San Raffaele hospital again on Sunday for what a spokesman said were routine checks.