Andy Burnham has said he is not making an immediate "commitment" to changing the tax-free personal allowance but added that it could be considered at the Budget.
The £12,570 threshold has been frozen since 2021, resulting in more people being dragged into paying income tax as earnings increase.
Burnham told the Times over the weekend that the "frustration about the personal allowance" he had heard while campaigning in the Makerfield by-election was "lodged in my mind".
The prime minister then told reporters on Monday that "all of this will be looked at" in the next Budget but conceded any change would be expensive and "difficult given the financial circumstances in which we find ourselves".
On Tuesday, people close to Burnham told BBC News that any changes to the personal allowance are not part of his initial plan to tackle the cost of living.
Asked whether he would make changes to the income tax allowance on Wednesday, Burnham said: "It's funny these days you can't answer a question honestly without people then reading everything else into it.
"I want to answer questions honestly in this job, as I said before. I think that's all about trust in politics and doing politics differently, and I'm determined to keep keep working in that way.
"It's just perhaps the way politics is these days and the way people report things, but that was an honest answer.
"There is no commitment at this point to change, but we will look at that at the Budget."
It comes after Burnham told his first cabinet meeting: "We've got to show fiscal discipline. We've got to show that our commitment to the fiscal rules is real and we're prepared to make difficult decisions in relation to that."
Burnham has committed to Labour's 2024 manifesto, which included a promise not to increase the basic, higher, or additional rates of income tax.
That would rule out increasing the top rate of income tax paid by higher earners to offset the cost of lifting the personal allowance.
The Institute for Fiscal Studies has estimated that unfreezing the threshold could end up costing £8.5-9bn a year.
Burnham said on Monday that he was "just kind of showing I have visibility of the issue and the impact it's had."
He told reporters it would be "looked at" in the Budget, but added: "Obviously it's difficult, given the financial circumstances in which we find ourselves."
