A poll published on Wednesday by RTR, the Romansh national broadcaster, showed that 45 percent of residents interviewed were opposed to the Olympic bid, while 42 percent were in favour.
The survey found nine percent were undecided and four percent were not interested.
The results mark a slide in support for the bid, budgeted to cost 2.8 billion francs, just two and a half weeks before a crucial cantonal referendum on the subject.
The survey was conducted by polling firm Demoscope on behalf of RTR, Radio SRF and two local newspapers.
A similar poll in November showed 43 percent of residents in Graubünden supported the Games, while 43 percent were opposed.
A nationwide poll last October indicated stronger support from Swiss residents as a whole, with 54.6 percent backing the Olympics bid.
The latest Demoscope poll found women tending to be more critical of the bid.
Concerns about costs have also arisen, with 57 percent of respondents fearing the Games would be too expensive, with cost overruns cited as a major worry.
When infrastructure expenditures are included, the total price tag could rise to above 4.5 billion francs.
The federal government has agreed to guarantee one billion francs of the costs, while backers anticipate 1.5 billion francs in revenues.
But a clear majority of those surveyed — 53 percent — also believe the Winter Olympics would boost the economy.
And Swiss President Ueli Maurer, who doubles as Sports MInister, has hailed the bid as an opportunity for Switzerland to “regain our good image” in the face of bad publicity linked to the country’s banks.
The survey involved interviews with 1,000 people with an accuracy rate of plus or minus 3.1 percent.