Labor should work with the Greens to tackle an “unfair, ineffective and inefficient” private health insurance industry, the Greens leader, Richard Di Natale, has said in a letter to the opposition leader, Bill Shorten.

“I believe that we should address these issues together and redirect part of the wasteful and ineffective private health insurance rebate to expand Medicare-funded dental care,” the letter, sent on Tuesday evening and seen by Guardian Australia, says. “A cooperative approach on this significant reform will increase the likelihood of its passage through the Senate.”

During his recent National Press Club address, Shorten accused private health insurers of “treating Australians like mugs, gouging people on the basis of a con”. Rising premiums combined with junk policies has seen complaints by consumers to the commonwealth ombudsman about private health insurance soar. Labor has proposed a short-term 2% cap on premiums to ease financial pressure on families. But in his letter, Di Natale told Shorten this was inadequate and would “only delay meaningful reform”.

On Wednesday night Di Natale, a medical doctor, will deliver a speech on private health insurance costs before an audience at the Sydney Institute. The private health insurance rebate will cost taxpayers $6.4bn this year.

“Far from taking pressure off the public system, the rebate in fact redirects a huge sum of money away from the public system and straight to the bottom line of private health insurance companies,” Di Natale’s speech, seen by Guardian Australia, says. “To put no finer point on it, the private health insurance rebate is the most significant con in the health system.”

“Even if it did work, what do we get for $7bn and rising every year?,” Di Natale will say. “An industry which has treated its customers appallingly, and that year on year sees premiums rising along with profits, while exclusions on policies grow.”

The Greens are calling for a single funding model for the health system similar to that proposed by the Australian Health and Hospitals Association (AHHA) for an Independent National Health Authority. In its submission to the Department of Health’s private health insurance review, the AHHA advocated for the removal or better targeting of the private health insurance rebate and called for the savings to be fully redirected into the public health system, including to broaden the list of items covered by Medicare.

“The Australian Greens would abolish the private health insurance rebate, and reinvest that money into service delivery in the public system,” Di Natale will say. “That is where the need is, and that is where the best value is. Let those who want to pay for private healthcare do so, but without the handout from the taxpayer.”

It comes as the federal health minister, Greg Hunt, on Tuesday ordered the Ombudsman to investigate the private health insurer Bupa. It follows a backlash from doctors to Bupa’s announcement that insured patients would qualify for gap cover only if they were treated in a Bupa-approved hospital or medical centre, and consumer anger that 720,000 Bupa policies will be downgraded. A health policy analyst with the Centre for Policy Development, Jennifer Doggett, said the Greens’ proposal reflected a large body of research which had found the private health insurance rebate was an inefficient and inadequate funding mechanism.

“It has led to rampant fee inflation in the private health sector, and this is why seeing a dental hygienist for a teeth clean typically costs more than seeing a GP,” Doggett told Guardian Australia. “The money currently spent on private health insurance subsidies would achieve much better outcomes and benefit many more people if it were used in the public sector.”

Doggett said that a single-payer system would allow for greater control over costs and reduce the inefficiency and fee inflation that results from having multiple, competing insurers.

“I think the key to making this work would be to ensure consumers have a central role in the Independent National Health Authority so that it was not captured by sectional interests, such as the pharmacy sector or medical peak groups,” she said.

“These interests have previously exerted a disproportionate influence on health funding and policy decisions to the extent that our system is really provider rather than consumer-driven. The Greens’ position puts pressure on Labor to declare a stronger position on the private health insurance rebate.”