The province of Tirol said in a statement on Sunday that 53.56 percent of the 312,237 valid votes cast across the region were against a bid to host the Games and 46.65 percent were in favour.
A total of 33,771 postal voting slips have been distributed and will be counted on Monday. Austrian media treated the result as definitive.
Innsbruck previously hosted the Games in 1964 and 1976.
The result, if confirmed, will be a blow to the International Olympic Committee (IOC) which has seen interest in hosting the summer and winter Olympics fall away in the last few years, scared off by the size, cost and complexity of the events.
Beijing and the nearby city of Zhangjiakou beat Kazakhstan's Almaty for the right to host the 2022 winter Games after the other competitors dropped out, including St Moritz/Davos and Munich whose bids were derailed in referendums.
Four of the six bidders pulled out of the race to host the 2024 summer Olympics, leaving just Los Angeles and Paris in the running.
In the end, the IOC voted to award the 2024 Games to Paris while Los Angeles was given hosting rights for the 2028 Games without a full bidding contest.
Rejection for the games was strong in Innsbruck itself where 67.41 voted against a bid.
(Writing by Brian Homewood; Editing by Christian Radnedge)
