Stirling Griff, a senator from the Centre Alliance party, told Australian Broadcasting Corporation (ABC) radio on Tuesday morning that the party was on the brink of announcing support for the package.
In return for the support of Griff and Rex Patrick, another Centre Alliance senator, the governing Liberal National Party Coalition will support measures that will lower the price of gas, reported Xinhua news agency.
"It's getting close ... we're 100 per cent behind income earners getting an extra boost in their pay packet," Griff told the ABC.
"We're broadly on board with the key proposition but we just need to cross the 't's and dot the 'i's. I'm confident we'll get there in the next day or two."
With the support of the Centre Alliance, the government will have 38 votes in favour of the package in the senate, only one short of the 39 required for a majority.
Whether or not the package is passed will likely come down to Tasmania's Jacqui Lambie, the only senator who has not declared a position on the policy.
Pauline Hanson, whose One Nation party holds two seats in the senate, on Tuesday, doubled down on her opposition to the cuts.
"These tax cuts are only going to help those people who are working," she told Nine Network television.
"Not the pensioners or those people on welfare who are not putting on their heaters in winter, the gas, because they can't afford the power."
The tax package, under which the average Australian would pocket an extra up to 1,080 AUD ($752.3) per year, was the foundation of Prime Minister Scott Morrison's successful campaign for re-election.
Mathias Cormann, the minister for finance, told reporters on Tuesday that he wants to see the package passed by the parliament before the sitting week ends on Thursday.
"We want to see this passed by the end of the week so obviously hard-working Austral can start to get their tax refunds into their bank accounts by the end of next week," he said.
Cormann again renewed calls for the opposition Australian Labor Party to "accept the verdict of the Australian people" in May's election by supporting the tax cut.
The ALP has announced support for the first two stages of the package, which will benefit low and middle income earners, but remains opposed to stage three, which will flatten the tax rate for all Austral earning between 45,000 AUD ($31,346) and 200,000 AUD ($139,316) to 30 per cent from 2024/25.
Instead, Labor leader Anthony Albanese said he wants to make the package "fairer" by splitting each stage into its own piece of legislation so that they can be voted on independently of one another.
"What we're going to do is fight for making this package better for the economy and also fairer," he told reporters on Tuesday.
"What we're talking about here is the government saying it's prepared to block tax cuts for workers now because of tax cuts that might happen, that it wants to happen, in 2025.
"Quite frankly, I believe that is an arrogant position on behalf of the Government."
( With inputs from IANS )