The Freedom Party candidate faces an independent, Alexander Van der Bellen, who has the backing of the Greens.
Mr Hofer topped the first vote but fell well short of an outright majority.
For the first time since World War Two, both the main centrist parties were knocked out in the first round, amid concerns over the migrant crisis.
Ninety-thousand people claimed asylum in Austria last year, equivalent to about 1% of the Austrian population, and the Freedom Party has run a campaign against immigration.
While the presidency is a largely ceremonial post, the president has powers to dismiss the government.
Austria is faced with a stark choice for its head of state: a Green Party professor, Alexander Van der Bellen, or Norbert Hofer of the far-right Freedom Party - a soft-spoken, charismatic gun enthusiast who won a decisive victory in the first round of voting in April.
For the first time since the Second World War, the traditional parties of the centre left and centre right were knocked out of the race.
Support for the Freedom Party has risen because of deep frustration with the established parties and, more recently, because of fears about the migrant crisis.
Rightwing parties are gaining strength in a number of EU countries. European leaders will be watching the result closely.
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In the first round, Mr Hofer secured 35% of the votes, while Mr Van der Bellen, polled 21%.
At his final election rally on Friday in Vienna, Mr Hofer, 45, sought to hammer home his message that immigrants needed to integrate.
"Those people who respect and love Austria and have found a new home here are warmly welcome," he said to applause.
"But those, it has to be said, those who do not value our country, who fight for Islamic State, or who rape women, I say to these people: this is not your homeland. You cannot stay in Austria."
The presidents of the European Commission and the European Parliament, Jean-Claude Juncker and Martin Schulz, have both expressed concern that Mr Hofer could win.
"I say to them very politely but firmly: we don't take orders from Brussels or Berlin," Mr Hofer said at the rally.
Mr Van der Bellen, 72, told his final rally in Vienna that it was likely to be a close race.
"I think it could be on a knife edge - fifty-fifty who will win, so this time, as with previous votes, but more than ever for this important election, every vote will count," he said.
At a news conference, he reflected: "As you know, I am 72 years old and I've experienced how Austria rose from the ruins of World War Two, caused by the madness of nationalism."
The two rivals had engaged in an angry TV debate earlier in the week, described as "political mud-wrestling" by commentators.
Such was the political shock at the far right's first-round win that the Chancellor (prime minister), Werner Faymann, resigned after losing the support of his Social Democratic party colleagues.
The Social Democrats and the People's Party have governed Austria for decades, either alone or in coalition.
At the last general election in 2013, they together won just enough votes to govern in a "grand coalition".
Incumbent President Heinz Fischer, 77, could not run again after two terms in office.
The European Union could see its first far-right president if Norbert Hofer wins the second, run-off round of the Austrian election.