Divisions have erupted over the sales pitch and design of the tax, with some demanding the Prime Minister accelerate the planned move from a fixed to a floating carbon price, or boost planned compensation for struggling families.

Julia Gillard is standing firm but her frontbench are in despair. “The carbon tax is killing us,” a senior minister told The Sunday Telegraph.

Cabinet tensions have arisen, with senior ministers accusing powerbroker Bill Shorten of leaking details of cabinet deliberations and shifting in his support for the PM.

Shorten denied both claims. “I support the Prime Minister,” he said.

One minister said: “But ministerial colleagues say he is regarded with huge suspicion: He was always seen by the Gillard camp as the weakest link. They regard him as their most suspect supporter.”

Gillard backer Dick Adams, a Tasmanian MP, has conceded “things are desperate”.

“We’re heading to a pretty heavy loss at the moment,” Adams said.

“Communication is a huge issue. Some people raise Swan. Can he communicate? We keep repeating our economy is the best in the world. But I don’t think people are talking about that down at the pub.”

There is little scope to scrap the carbon tax: it is now legislated, and was part of the minority government’s crucial deal with the Greens.

It comes into effect at a fixed price on July 1 and will become a floating-price emissions trading scheme in mid-2015. The government could ease its impact now by offering families more compensation or bringing forward the ETS start date, but this would smash Wayne Swan’s planned surplus, which partly relies on revenue from the fixed carbon price for the next three years.

Cabinet ministers are in despair over the carbon tax. Another minister said the real mistake was not going to an early election in 2010 over the previous ETS. “It’s a huge issue out there. There’s been the hope that things will turn around. I am not sure the compensation package will do that,” the MP said.

The government is already focus-testing messages for its multi-million dollar ad campaign on the planned compensation deal for families.

One of the men who helped install Gillard told The Sunday Telegraph the Prime Minister was “delusional”.

She won the February 27 leadership ballot by 71 votes to 31, including strong support from the powerful NSW Right. Six of the faction’s MPs voted for Kevin Rudd and a dozen for Gillard, but that support is now believed to have collapsed. Some key NSW Right figures, including Tony Burke, still support the PM, but she has also lost the backing of key Left faction leaders.

Calling for Combet’s name to be included as a possible PM or deputy PM, one Left powerbroker who voted for Gillard in February is urging her to quit.

“They should hand in their commission. Both of them. Swan and Gillard. Because they can’t box their way out of it,” he said. “My own view is we should skip a generation. The next election isn’t going to be about policy. It’s going to be about ethics, morality and integrity. I like Greg Combet for that reason.”

But the MP said he had no discussions with Combet over the idea, whom he described as notoriously reticent to get blood on his hands.

Another Labor minister said, if Gillard was to resign, Combet should be considered, despite his role as an architect of the carbon tax.

“If the leadership becomes vacant Greg should be on that list. If Bill Shorten is an option, Combet certainly is,” the minister said. “The week of budget, and after, will be critical. I think people are increasingly despairing about the chances for this ending in anything other than a disaster.”