In Question Time on Wednesday, Mr Hawke told parliament it "wasn't good enough" for Labor to now support the bill so close to an election "when you know we have run out of time, potentially, to pass the bill".
Labor immigration spokeswoman Kristina Keneally says the government needs to decide if they want a "fight or a fix" on the bill.
"Labor will not oppose this bill in the House of Representatives. We assume there is going to need to be some amendments from the government in the Senate," she told Sky News on Wednesday.
"We are hopeful (the government) will continue to work with us on some of the unintended consequences."
Senator Keneally said the immigration minister had invited her in writing to a negotiation on the issue, but when she responded she did not hear from him again.
The government's proposed bill would give the immigration minister further powers to cancel visas and deport people.
The proposal seeks to "strengthen the character test" by allowing the government to cancel the visas of non-citizens if they have committed an offence punishable by a maximum prison sentence of two years.
Visas could also be cancelled if non-citizens were deemed a risk to the broader community.
Mr Hawke's office confirmed there would be no negotiation and the government would not entertain amendments.
Prime Minister Scott Morrison says the changes are needed to close loopholes in existing laws.
"There are foreign citizens who have been convicted of very serious crimes against Australians, who have been jailed, and as a result of the law as it stands today ... we are unable to deport," he told parliament.
But opponents of the bill say if passed, the new laws would lead to more people being detained, including people who arrived in Australia as children and have lived almost all their lives here.
"The bill would lower an already excessively low threshold for the minister to be able to rip a person away from their families, lock them up in a detention centre and deport them to a country that is not their home, even when that person has lived in Australia for decades," Human Rights Law Centre senior lawyer Scott Cosgriff said.
The Asylum Seeker Resource Centre estimates the bill will increase visa cancellations fivefold - after an existing increase of more than 1000 per cent since 2014 - and disproportionately affect refugees the most.
ASRC spokeswoman Jana Favero called it a racist law.
"It explicitly subjects non-citizens and refugees to a separate legal system, where some serve light sentences of community service and others are indefinitely detained for exactly the same offence," she said.
The legislation was brought to parliament in October 2021, but was defeated in the Senate.
Labor and the Greens argued at the time the changes would mean visa holders charged with minor offences would be deported.
New Zealand has long-standing concerns about Australia's deportation policy and opposes any strengthening of it.
