Personal income tax cuts will soon pass the Senate after breezing through the lower house with the backing of Labor.

The tax office says workers will start receiving the tax relief as soon as practicable.

Labor has thrown its support behind the bringing the cuts forward by two years, in a bill which also includes tax concessions for small business and more instant asset write-offs.

“We are prepared to vote to get these through the house today and into the Senate as soon as possible,” shadow treasurer Jim Chalmers told parliament on Thursday.

“We do acknowledge that these are serious times, we are in the deepest, most damaging recession for almost 100 years.”

But the Greens says the money to implement the tax cuts could be better spent to stimulate the economy.

“Right now today, the government and Labor are funnelling tens of billions of dollars out the door that are going to go the pockets of big corporation, and are going to go to the millionaires,” Greens leader Adam Bandt told parliament.

Dr Chalmers wrote to tax commissioner Chris Jordan on Wednesday to confirm Labor’s support for the tax cuts, which are backdated to July.

There is a longstanding practice that tax measures are implemented as soon as bipartisan support is confirmed.

“The ATO will work closely with providers of payroll software and employers to ensure the reduced withholding associated with the threshold changes and the increase of LITO (low income tax offset) is reflected in software as soon as practicable,” the agency said.

The top threshold of the 19 per cent personal income tax bracket will increase from $37,000 to $45,000 backdated to July 1, while the top threshold of the 32.5 per cent income tax bracket will increase from $90,000 to $120,000.

At the same time, the low income tax offset will increase from $445 to $700.

Adjustments will be made to ensure taxpayers have the correct amount of tax withheld from their pay going forward.

“Any ‘over-withholding’ that occurred prior to the employer updating their payroll software and processes will be included in the tax assessment of the employee at the end of the income year,” the ATO said.

The low and middle income tax offset will be retained for the 2020/21 financial year and will be received after individuals lodge their tax return for the year.