Launched at the end of September, the initiative ‘Health insurance: For the organisational freedom of cantons', wants to allow each canton to create an institution that can fix health insurance premiums at a set rate within that canton.
Private insurers would still exist and administer health insurance, but premiums for basic health insurance (LaMal) would be the same for every resident of that canton.
And according to a survey by online comparison site bonus.ch, published on Tuesday, over 64 percent are in favour of the plan.
People in French-speaking areas were most in favour, with 71.4 percent judging the idea ‘good' or ‘very good', said bonus.ch in a statement. A majority of those in Italian-speaking Ticino also supported the plan (65.4 percent). However people in Swiss German parts were divided, with only 51.5 percent ticking ‘good' or ‘very good'.
If only counting those who responded with ‘very good', people in the canton of Jura were most enthusiastic, at 56.5 percent. That was followed by the cantons of Geneva, Vaud and Fribourg.
Swiss German cantons were the least enthusiastic, with only 15.9 percent of respondents in the canton of Zurich ticking ‘very good'.
Men were slightly more in favour than women, found the survey, however age had no bearing on the results.
The committee behind the initiative is backed by four politicians who are the health ministers within their cantonal governments.
They now have until April 2019 to collect the required 100,000 signatures to push it to a referendum.
If the Swiss people were to vote in favour of the plan it would be a massive change for a system that many feel is unfair, unwieldy and too expensive.
Currently, residents must choose from one of over 60 private health insurers for their compulsory basic medical insurance.
Though the cover is the same, premiums vary wildly.
Since the law on LaMal came into effect in 1996, the average standard premium has risen by 4.6 percent a year.
Premiums for 2018 will rise by an average of four percent, the government announced at the end of September, citing rising health care costs as the cause.
In 2014 Swiss voters rejected a plan to scrap the current system and create a single, publicly-run health insurance scheme.