Speaking to the press including Le Temps, Carole-Anne Kast, president of Geneva's Socialist Party, said they would prefer action to be taken at federal level but that they were not willing to wait any longer.
“We are pleading to Bern for reform but we must take action locally now. We can't wait,” she said.
The initiative doesn't aim to help people who are already receiving social assistance to pay their premiums, but those working people who pay more than ten percent of their income on health insurance.
Campaigners have until March 2018 to gather the required 5,100 signatures to push it to a cantonal referendum.
Geneva residents pay among the highest premiums in Switzerland for their compulsory basic health insurance (LaMal), which provides the same cover wherever you live in the country but is calculated on the costs to insurers of providing healthcare in that canton.
According to figures from the Swiss statistics office quoted by Le Temps, the average monthly premium in 2017 (based on a 300 franc excess with accident cover) was 447 francs in Switzerland as a whole, but 554 francs in Geneva, set to rise to 583 in 2018.
Premiums will rise by an average of four percent in 2018, but again that varies, with the canton of Vaud seeing the highest increases (6.4 percent) against Schwyz on the lowest (1.6 percent).
In 2016 the Swiss upper house of parliament rejected a similar initiative to cap premiums at federal level, saying it should be up to the cantons to legislate on the issue.
Premiums for compulsory medical insurance are set by the country's 60-odd private insurers each year, usually at the end of September, with figures approved by the federal government.
In 2014 voters rejected an initiative to change the system to a single, publicly-run insurance scheme which backers said would have helped to reign in premiums.