More than 200 MPs have signed a letter to the prime minister calling for government promises to Windrush migrants to be written into law.
The letter, coordinated by Labour MP David Lammy, said concerns over compensation, housing and legal rights had not been settled.
The Home Office said Amber Rudd would speak in Parliament on Monday.
The home secretary is accused in the letter of making up immigration policy “on the hoof” to diffuse the situation.
The letter, addressed to Theresa May, said any promises made by the government in response to the Windrush crisis should be enshrined in law “without delay”.
It continued: “We are calling on you to do this by bringing a statutory instrument before Parliament to ensure that that the measures are implemented as quickly as possible.”
The signatories are predominantly Labour MPs but politicians from the SNP, Plaid Cymru and the Green Party – as well as one Conservative – also feature.
The Windrush row erupted after it emerged some migrants from Commonwealth Caribbean countries who settled in the UK from the late 1940s to the 1970s, and their relatives, had been declared illegal immigrants.
Some of the Windrush generation have been threatened with deportation, lost their jobs, or been refused access to medical treatment.
In a separate letter, Labour MP and shadow home secretary Diane Abbott has called for a full inquiry into whether the home secretary had breached the ministerial code over government immigration removal targets.
Ms Abbott said Ms Rudd’s explanations as to why she had told MPs there were no such targets when in fact there were “stretch credulity to the limit”.
Downing Street has said there is no need for an investigation because ministerial code only requires a resignation if a minister knowingly misleads parliament – a charge the home secretary denies.
Ms Rudd tweeted on Friday night admitting she had been copied in on a memo that mentioned migrant deportation targets but insisted that she had not actually read it.
Communities Secretary Sajid Javid, meanwhile, has appealed to ethnic minority voters not to abandon the Tories in this week’s local elections over the Windrush scandal.
Mr Javid, whose parents emigrated from Pakistan in the 1960s, said his first reaction when he heard people were being wrongly threatened with deportation was that it could have been his family.
However, in an interview with the Sunday Telegraph, he said the government was committed to “put things right” and he urged ethnic minority voters to look at the “bigger picture” when it came to Thursday’s vote.