The opposition wants to wait and see but has reservations about it unless certain inflation offsets can be part of the equation.
The irony is that the two may well end up adopting each other's original position on what has become a defining cost-of-living issue.
The government's three-month cut to the fuel excise kicked in at the start of April, resulting in the cost of petrol and diesel being slashed by 26.3 cents a litre in response to soaring prices caused by the US-led war on Iran.
The prime minister left the door ajar for an extension on Saturday despite advising Australia now holds a healthier supply of fuel than it did before the conflict erupted in late February.
"We'll make an assessment in the lead up to July 1," Mr Albanese said in Sydney.
He went on to spruik other forms of tax relief commencing at the start of the new financial year that may cushion the blow should the excise cut be allowed to lapse.
"On July 1, of course, we've got our first tax cuts, one of the five instalments of tax cuts that have been put in place by my government," he said.
Australia now has 44 days' worth of petrol, two days more than last week's update and eight days more than the day the Iran bombing began.
The nation has 36 days worth of diesel and 35 days worth of jet fuel.
Asked whether the opposition would support extending the excise cut, Shadow Treasurer Tim Wilson said he'd prefer to hold off commenting until Australia's fuel supply challenge pans out.
"We're waiting to see both fuel supply and the global context before we provide an answer," he told reporters at Melbourne Airport.
"When we led this conversation originally and put the proposal forward, we made sure there were inflationary offsets because what we didn't want was more debt petrol on the inflation fire."
The prime minister says Australia is "in a better position than anyone was predicting prior to Easter".
The nation remains at level two of the government's fuel plan that requests users only buy what they need and take voluntary steps to use less.
The federal budget included a multi-billion-dollar fuel resilience package, including a $7.5 billion fuel and fertiliser security facility and $3.2 billion Australian fuel security reserve.
It is designed to facilitate at least 50 days of onshore fuel supply and storage of diesel and avgas.
Australia has also been ramping up efforts to secure supplies, with an additional three spot-market diesel cargoes secured on Friday.
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