Austria's far-right presidential candidate Norbert Hofer has said he would push for a referendum on EU membership, if the EU became more centralised after Brexit.
He told the BBC that while the 28-country bloc was important for Austria, he wanted a "better European Union".
Austria's presidential vote re-run is on 4 December and the polls indicate the result is too close to call.
If Mr Hofer wins, he will become the EU's first far-right head of state.
His independent rival, Alexander Van der Bellen, won the election in May but the vote was overturned.
The former Green party leader had beaten Mr Hofer by a margin of only 30,863 votes and Austria's high court backed a complaint that election rules on postal voting had been broken.
Mr Hofer's Freedom Party had argued that postal ballots were illegally and improperly handled in 94 out of 117 districts.
Read more on Austria's neck-and-neck vote:
Although the role is largely ceremonial, the Austrian president does have the power to dissolve the lower house of parliament, the National Council, and trigger elections.
The Freedom Party candidate told the BBC that no-one should be scared of him as a leader. However, while Muslims living in Austria were part of the country, "Islam is not part of our values", he said.
The far-right leader has insisted that a vote for him would not be a vote for Austria pulling out of the EU, or "Oexit".
However, Mr Hofer told the BBC that two issues might change his mind.
One was any future decision to admit Turkey into the EU. Although Turkey is in accession talks, there seems little likelihood of it joining the EU any time soon.
And then there was the EU's response to the UK leaving the bloc.
"If the answer to Brexit would be to make a centralised European Union, where the national parliaments are disempowered and where the union is governed like a state. In this case, we would have to hold a referendum in Austria, because it would lead to a constitutional change," he said.
How far right is Hofer?
- Norbert Hofer is a member of a pan-German nationalist fraternity called Marko-Germania
- Some of his close colleagues are from an extremist background
- One is Freedom Party leader Heinz-Christian Strache
- The Freedom Party was founded in 1955 by a former general in the Nazi SS
- The cornflower that party members wear was used by pan-German nationalists in the 19th Century but also as a secret symbol by the Nazis
- Mr Hofer wants to reopen debate over self-determination in South Tyrol, annexed by Italy in 1919
Vying to lead Austria
- Age: 45
- Background: Aeronautical engineer
- Politics: Far-right Freedom Party
- On Trump: "I'm no Trump fan, but I do have high hopes of improved relations with Russia."
- On Germany's decision to open its doors to refugees: "I say we're not managing. It was a major mistake that has inflicted massive costs on the entire EU, as well as Austria and taxpayers."
Alexander Van der Bellen
- Age: 72
- Background: Economics professor
- Politics: Former Green Party leader
- On Trump: "His campaign style, the sexist attacks were unacceptable. As regards content he has raised concerns - let's give him several months"
- On immigration: "I want a liberal, cosmopolitan Austria. What I do object to is lifting the old borders"