National party senators are joining the Labor party and the Greens in questioning the cuts to single income families’ benefits that the Abbott government says are essential to pay for the new childcare package that forms the centrepiece of its budget.

On Sunday the government announced more generous childcare payments for all working families, regardless of income, but insisted the new payments depended on the passage of cuts unsuccessfully proposed in last year’s budget to existing payments for single income families.

Labor’s shadow treasurer, Chris Bowen, said that linking the extra childcare assistance to cuts to the existing payments for low income single income families was “disingenuous and cruel” and the Greens were also critical.

But Coalition backbencher, Queensland Liberal National party senator Matthew Canavan agrees the payments should not be linked.

“We are deeply concerned about the divide between support for working families and support for families where one parent stays at home … our system massively penalises families where a parent stays home to look after children and we think it does not properly value the benefits of unpaid work,” he said.

Canavan and fellow Nationals, including senators Bridget McKenzie, Barry O’Sullivan and John Williams have previously demanded the government abandon the cuts to family tax benefit B (paid to single income families) which are worth almost $2bn over five years and which the prime minister, Tony Abbott, and the treasurer, Joe Hockey, say are essential to pay for their new childcare package.

The Nationals senators want families earning one income to be allowed to split those earnings between two parents for tax purposes, in order to claim the tax-free threshold twice – something that would cost the budget $1.5bn a year.

And with the government setting up a Senate showdown with Labor and the Greens over the family tax benefit savings, Canavan did not rule out crossing the floor to vote against them.

“I’m not ruling anything in or out,” he said. “We’re not going to get walked over, but I don’t take that kind of decision lightly either.”

“I’ll have my say in the party room, but there is a very, very strong level of support for the idea that we should be looking after single income families,” he said.

The package was already hitting resistance in the Senate, with Labor insisting it will not back the cuts to family tax benefits that the government says are the only way to pay for the new childcare plan.

“Any package which assists with cost of living and workforce participation, making it easier for parents to participate in the workforce, is welcome, and we’ll have a look at that, but for the government to link this to cuts to family payments is disingenuous and cruel,” Bowen, told the ABC’s Insiders program.

Greens senator Sarah Hanson-Young said the tactic of linking the childcare payments to the family tax benefit cuts amounted to “blackmail” and low-income families needed childcare to look for a job.

“If they really want mums to be able to ease themselves back into the workforce they’ve got to be given those guaranteed childcare support places up front,” she said.

“How do you go to a job interview if you’ve got nowhere to put your child while you’re there?”

The government will spend $3.5bn over the next four years on the childcare package – which takes effect after the next election in July 2017 – in addition to the about $7bn a year already budgeted for childcare spending. The new system will:

• offer more generous payments of 85% of the cost of care to all families earning up to $65,000;

• remove the $7,500 a year each child cap on payments to all families earning up to $185,000 a year;

• continue to offer the 50% rebate to families earning over $185,000 and increase the annual cap for each child for these families to $10,000.

But to save money it will also:

• remove all childcare subsidies for families earning more than $65,000 where both parents are not in the workforce, replacing them with a sliding scale of payments to encourage parents to increase their hours of casual or part-time work;

• reduce the number of hours of subsidised childcare offered to non-working families earning under $65,000 to 12 hours a week, but continue to subsidise those hours recognising that children from these families may have particular need of the pre-school education that childcare provides;

• Stop parents from “double-dipping” by accessing both government- and employer-funded paid parental leave.

And the entire package depends upon the Senate passing the cuts to family tax benefits proposed in last year’s budget but rejected by the Senate. They included:

• ending family tax benefit B (paid to single-income families) when the youngest child turns six, saving $1.9bn over five years;

• freezing all family tax payments for two years, saving $2.6bn over four years;

• cutting end-of-year family tax benefit supplements, saving $1.2bn over four years.