During Prime Minister's questions, opposition leader Jeremy Corbyn asked Mrs May to explain why she did not act on internal government memos warning about potential discrimination caused by the policies, which are the focus of a massive political scandal.
Amber Rudd is set to face a grilling from MPs over her handling of the Windrush scandal as Labour stepped up its calls for the Home Secretary to resign. "Isn't it time she took responsibility and resigned?"
The PM said Windrush families would get the support they needed but the public wanted action on illegal immigration.
After noting that May had quoted her on immigration previously, Cooper said: "Let me say to the PM, do not try to hide behind me or the Labour Party because she was warned repeatedly of the damage her obsession with her net migration target was doing".
Urging the home secretary to quit, Mr Corbyn said Ms Rudd had "inherited a cruel policy" when she succeeded Mrs May in 2016 and had made it even tougher.
"Do not try to hide behind the cabinet when they don't agree with you on this and are trying to clear up the mess. Because a few years ago the prime minister said: 'I'm actually sick and exhausted of a government minister who simply blames other people when something goes wrong.' What's changed?"
May has refused to respond to calls to sack her Home Secretary Amber Rudd.
It also emerged before PMQs in the Commons that Foreign Secretary Boris Johnson, a key figure in the race to replace May, favours an immigration amnesty for all long-standing Commonwealth migrants and had championed that idea at this week's Cabinet meeting.
Prime Minister Theresa May has said the "Windrush generation", people from the Caribbean who came to Britain as children after World War Two, were British and her government would not tell them to leave the country, writesElizabeth Piper.
"Nobody's trying to blame anybody else", she said. "We remain vigilant on the issue such that our Mission in the United Kingdom, with whom the Ministry is in regular communication, are monitoring and maintaining contact with the Home Office on the issue", added Johnson Smith.
"These policies swept up British citizens and legal migrants causing them enormous suffering, as she was warned", he told MPs.
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Now if that isn’t the epitome of what a BFF is, I don’t know what is! Selena continues to be on chronic medication. There is also a huge list of celebrities past like famous painter vincent Van Gogh who suffered with this disease.