Van der Bellen (72) is running against 45-year-old Norbert Hofer of the popular anti-immigration Freedom Party (FPÖ).
The presidential election re-run will take place on December 4th, after a scandal-ridden race that could usher in the European Union's first far-right head of state.
Austria’s high court annulled the first election result from May after upholding the narrowly defeated FPÖ's claims of procedural irregularities.
The Kurier newspaper reports that Van der Bellen was placed under permanent police protection on Wednesday morning. Counter-terrorism police have been creating a plan to secure his apartment and the routes which he regularly travels. Up to three armed officers will accompany him throughout the day.
Van der Bellen has been receiving death threats via Twitter and Facebook. One Twitter user who claims to be based in Vienna said he planned to “shoot so many holes in Van der Bellen that he will look like a golf course”. He also tweeted pictures of a swastika and guns.
Insiders say that the threats against Van der Bellen have increased in recent weeks and that many of the comments are being posted in known right-wing websites.
Police in Vienna are investigating the tweets and have applied to have some of the Twitter accounts blocked. The Interior Ministry says that is is taking all the death threats very seriously.
In May, before the first election, Van der Bellen also needed police protection – but this was later scaled down. At that time he was receiving threats on a regular basis, particularly via the Facebook page of FPÖ leader Hans-Christian Strache, after he told German media that he thought Strache would be incapable of forming a government should he win the next parliamentary election.
Norbert Hofer is currently not in need of police protection.