Jeremy Corbyn has called for the home secretary, Amber Rudd, to resign over a "cruel and misdirected" immigration policy that the Labour leader said was responsible for the hardships faced by the Windrush generation.
The prime minister said a Labour government would not be "kind or fair to anybody".
The row came amid deepening anger at the way members of the Windrush generation, who arrived from the Commonwealth in the decades following the Second World War and who have now been threatened with deportation, have been treated.
The issue "was covered by newspapers, and MPs bringing it forward anecdotally over the past three or four months, and I became aware that there was a potential issue", she told parliament's home affairs committee, according to the Guardian.
"At last she's been forced to act upon it", Corbyn said. "I didn't see it as a systemic issue until very recently", she said.
The revelations have prompted fresh criticism of the Government's "hostile environment" policy to tackle illegal immigration.
But she said a distinction should be drawn between those people who had settled in the United Kingdom legally and contributed to British life and those with no right to be in the UK.
After then being forced to sit through a question from Conservative Andrea Jenkyns, who took her husband Ed Balls' seat in Morley, she finally caught the Speaker's eye and did not disappoint. "The problem here is that people were not properly documented".
"But for governments of every colour, including that in which the Rt Hon Lady served, action has been taken against illegal immigrants".
The Windrush generation, named after one of the first ships that arrived from Jamaica in 1948, are citizens of the United Kingdom and colonies, a type of citizenship conferred by the British Nationality Act of 1948.
Those who have been living legally in the United Kingdom for decades have lost their jobs, been denied access to NHS treatment, benefits and pensions, had their driving licences withdrawn and been warned they face deportation. However, although many have taken United Kingdom citizenship or have documents to prove their status, some do not. "Last week, the current home secretary admitted the Home Office sometimes loses sight of the individual".
May responded by saying Rudd would "set out the details of that compensation scheme in due course".
In another step, the Government published guidance for employers and landlords to clarify that people in the Windrush generation are legally able to rent property and work in the UK.