The mayors of both venues last month announced they were planning a twin bid to stage the Games, but said they were not considering a referendum after people in the state of Tyrol voted against Innsbruck entering the race in October.
However, Strache warned against any decision to force through a candidacy without the approval of the population.
"There has to be a desire that is largely shared by the region and the people who live there. One cant go against the citizens," said Austria's deputy chancellor, who is also leader of the far-right Freedom Party.
"Such a project must always be supported by the population," Strache said in an interview with APA news agency, without explicitly calling for another referendum.
An ever-shrinking pool of cities willing to take on the multi-billion dollar cost of staging the Games saw Beijing beat only one rival bidder, Almaty, to win the right to host the 2022 showpiece.
Promoters of a joint Graz-Schladming bid insist almost all the required infrastructure to stage the Olympics is already in place. Schladming, which hosts a floodlit World Cup slalom each year that draws huge crowds, was the venue for the 2013 world ski championships.
Austria, renowned for its winter sports prowess, has twice hosted the Games, at Innsbruck in 1964 and 1976.
Venues then chosen to make an official bid will be announced in the autumn with the winner revealed in September 2019.
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