An Isopublic survey, commissioned by the SonntagsBlick newspaper, showed 54.6 percent in favour of the candidacy for the Games envisioned for St. Moritz and Davos.
The poll of 1,033 Swiss residents was conducted between October 17th and 20th.
Those backing the games appear to be unfazed by the 2.8-billion-franc cost ($3 billion) of staging the sporting event as estimated by a Graubünden organizing association.
The federal government last month agreed to contribute 30 million francs for the bid.
The cabinet is also committed to covering at least one billion francs of the costs, while backers anticipate 1.5 billion francs in revenues.
However, infrastructure costs of 1.7 billion francs, coupled with security expenses bring the potential total bill to around 4.5 billion francs.
The Isopublic survey showed strong support for the Olympic candidacy from young people, with three-quarters of those under the age of 35 saying they favoured the project.
Results showed backing from around 60 percent of residents in the Alpine areas of Graubünden and Valais and the foothills of the Alps.
Voters in Graubünden are set to decide whether to approve the bid in a referendum set for March 3rd 2013.
Sports Minister Ueli Maurer has said the Games offer a “great opportunity for Switzerland”.
The country has not hosted the Winter Olympics since 1948, when the Games were held in St. Moritz.
The mountain resort also hosted the Games in 1928.
But bids by Switzerland to stage the 1976, 2002 and 2006 Winter Olympics (all in the canton of Valais) were unsuccessful.