The government hopes Parliament will find the time to pass the HIA bill before it is dissolved, a Downing Street spokesperson has said.
On Thursday, the House of Lords finished its work on the bill enabling compensation payments to be made to victims of institutional abuse in NI.
However, it is not clear if there is time for it to go back to the Commons.
Victims have lobbied for compensation since the Historical Institutional Abuse (HIA) Inquiry ended in 2017.
A Downing Street statement said the government was committed to ensuring victims got the "redress they deserve".
"That is why it was one of the first bills introduced following the Queen's Speech," it continued.
It added that "given the importance and sensitivity of the bill" the government hoped it will be scrutinised and passed before Parliament is dissolved on Wednesday.
In the House of Lords, the Liberal Democrats said it would be on the government's head if the legislation is not approved in time.
"It really is something the government would not only be well advised to find time for, but should recognise that there will be absolutely no understanding for the incapacity to find the few minutes that would be needed," said Lord Bruce.
There is cross-party support for the bill. However, it would need to pass by next Tuesday night.
During prime minister's questions on Wednesday, the prime minister stopped short of making a commitment to pass the legislation before Parliament is dissolved.
Instead Boris Johnson told MPs that the "most powerful way" to address the issue would be to get Stormont up and running again.
Jacob Rees-Mogg, who is the leader of the House of Commons, was also pressed on the issue by MPs on Wednesday night.
He said it would not happen on Thursday and there could be an issue getting Royal Assent for the bill on Monday.
Earlier, former NI Secretary Lord Hain criticised Mr Rees-Mogg's handling of the bill.
Lord Hain said there is "no excuse" and that it could be passed through in under an hour.
"Listening to Jacob Rees-Mogg, it's just not good enough," he said.
"It could have gone into the Commons this evening.
"For Jacob Rees-Mogg to say it wasn't in the draft of his statement, well what is the leader of the Commons for?"
Parliament will be dissolved on Wednesday for the general election.
Prime minister letter
Campaign group Survivors and Victims of Institutional Abuse (Savia) wrote to Prime Minister Boris Johnson on Wednesday to "beg that you ensure" the legislation is passed before Parliament is dissolved.
The letter said Mr Johnson had made a commitment to introduce the legislation during the recent Queen's Speech and urged him to "please do the right thing".
Marty Adams from the group Survivors Together told BBC News NI's Good Morning Ulster programme that "victims cannot take any more".
"I don't think the prime minister really understands what he's doing to victims," he said.
The HIA inquiry, chaired by the late Sir Anthony Hart, investigated historical allegations of child abuse in residential institutions run by religious, charitable and state organisations.
It examined 22 institutions and its remit covered a 73-year period ranging from 1995 back to the foundation of Northern Ireland.
The inquiry's final report in January 2017 recommended that all survivors of institutional abuse receive tax-free, lump sum payments ranging from £7,500 to £100,000.
However, it was published just a few days after the collapse of Northern Ireland's devolved government, and no ministers were in post to set up the redress scheme.
Since then victims' groups have lobbied the Northern Ireland Office to pass the necessary legislation through Westminster and brought a judicial review challenging the lack of government action.