The coalition has no plans to outbid Labor on income tax cuts, with Treasurer Scott Morrison saying the government will stick to its seven-year plan.
Bill Shorten used his budget reply speech on Thursday night to announce support for the government's planned $530 offset for low- and middle-income earners - and then go further.
Labor's $5.8 billion plan will give $350 a year refunds to people earning $25,000, rising to $928 a year for people earning up to $90,000.
Above that, the extra refund tapers down to $140 for people earning $120,000 a year.
But Mr Morrison rejected suggestions he could match Labor's cuts.
"We carefully considered our plan," the treasurer told 3AW on Friday.
Mr Shorten said five upcoming by-elections, triggered by the dual citizenship saga, will give voters a choice between the coalition's corporate tax cuts and Labor's proposal for greater income tax relief.
"They are an early opportunity for Australians to judge, to make an early referendum, on what we saw in Tuesday night's budget," he told reporters in Canberra.
"If you want to give massive tax cuts to multinationals and big banks, vote for the Liberals and Malcolm Turnbull.
"If you want to see 10 million working Australians get better tax cuts - $928 per year for many of those Australians - if you want to get a better deal for working and middle class Australians, vote for us."
Finance Minister Mathias Cormann labelled Mr Shorten a "dishonest populist" and claimed his numbers didn't add up.
"They're spending the same money several times over," Senator Cormann told Sky News.
Prime Minister Malcolm Turnbull said Australians could not believe "Unbelieva-Bill".
"He has not produced any of these costings - it is perfectly obvious he has done these numbers on the back of an envelope following Scott Morrison's budget speech on Tuesday night," Mr Turnbull told reporters in Sydney.
Asked whether his numbers were rubbery, Mr Shorten said the government was jealous.
"It's very straightforward. The numbers speak for themselves," he said.
