A privatised Medibank Private would not greatly change the way it does business, experts say, and its listed rival says it would welcome the competition.
But the ownership of Medibank rests on the results of the federal election. Labor wants to retain Medibank as a government-owned enterprise but the Coalition's policy is to sell it.
The Greens do not have a position on a sale, but its health spokesman, Richard Di Natale, has said he would consider supporting one if the proceeds went to public health.
Carol Bennett, head of the Consumers Health Forum, said a privatisation made sense. "It's a private health insurer, it should probably be in private hands," she said.
NIB, the only listed health insurer, said a market listing of the country's largest insurer would "bring a lot of attention and focus to the industry, which is good for us".
"You can't throw a rock that big in the pond and not expect some ripples," it said.
Medibank has 3.8 million members. It also has a contract to service the Australian Defence Force and has branched into life insurance and pet insurance.
It has yet to release its annual report for the 2013 financial year, which will confirm how much it has paid the government in dividends over the 12 months to June.
But its managing director, George Savvides, told Senate estimates last month that Medibank had a two-year "fairly intense program of cost reduction".
Grahame Danaher, managing director of the not-for-profit health fund Westfund, said the comments suggested Medibank was preparing for privatisation.
It has been widely suggested that Medibank has become more aggressive recently in its commercial approach. It has appointed two new board members since March, Peter Hodgett and Elizabeth Alexander.
In the 2012 year, it reported a 57 per cent slump in net profit to $126.6 million, due to higher claims costs and a sharp fall in investment income. Revenue reached a record high for a second year running at $5.4 billion.
NIB shares rose 5 per cent through 2012, valuing the company at $926 million.
But the newly listed comparison site iSelect, which derives most of its revenues from health insurance sales, closed at $1.62 on Friday - well below its listing price of $1.85.