According to the survey, the FPÖ would win some 35 percent of the vote, just one point short of the SPÖ, which has controlled the Austrian capital since 1945.
The FPÖ, led by 46-year-old Heinz-Christian Strache, has gained steadily in the polls since the spring, thanks in part to the migrant crisis, like other far-right parties across Europe.
In recent months Austria has become a major transit country for tens of thousands of migrants entering from Hungary -- having travelled up through the western Balkans -- bound for northern Europe, in particular Germany.
Late last month the FPÖ doubled its score in regional elections in Upper Austria, winning 30 percent of the vote for a second place behind the conservative People's Party (ÖVP).
In June, Strache's party joined in an improbable coalition with the Social Democrats in eastern Burgenland after winning 15 percent of the traditionally left-wing province.
Analysts say the mainly working-class electorate of the SPÖ has partly shifted to the FPÖ.
The survey by polling firm Market for the Standard daily found the FPÖ up nine points while the SPÖ was down eight.
If the Freedom Party wins in Vienna, it would be a symbolic political earthquake, but the party would not necessarily take power in the capital, since runners-up would likely form a coalition.
The Austrian capital has long prided itself on its leftwing credentials, going by the nickname "Red Vienna".