MPs have said the targets could have led to enforcement officers to target "low-hanging fruit" - people living in the United Kingdom legally but without the correct documents, like numerous "Windrush generation" - a term referring to people invited to Britain to plug labour shortfalls between 1948 and 1971, some of whom have since been labelled as illegal immigrants.
Ms Cooper said she had asked the Home Office whether it has targets for immigration removals after "contradictory evidence" from the Home Secretary and the Immigration Services Union.
A 2015 report that came to light yesterday showed the Home Office did set targets for voluntary departures of people who could not lawfully stay in the UK. "Isn't it time that the home secretary considered her honour and resigned?"
In response to an urgent question from Labour in the House of Commons on Thursday, she said she did not know about the targets.
"When that Prime Minister was Home Secretary, meeting those targets would have been made a lot easier by describing people who had every right to be here as not having the right to be here".
Some of the Windrush migrants have been made homeless, lost their jobs, been threatened with deportation and denied benefits, according to the Guardian newspaper.
"We have let you down and I am deeply sorry", she said.
Lucy Moreton, general secretary of the Immigration Service Union, had told the MPs a national target, broken down regionally, had been set to remove people in the United Kingdom illegally, and staff were under "increasing pressure" to meet it.
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There was also a target to remove families in the country illegally, though the inspection report said it was not a "useful performance measure".
"I understand the House will want to hold me to account for that, but I am confident the changes I am committed to putting in place and the transparency that will go with that will deliver the changes that are expected".
In her evidence to the committee, Ms Rudd rejected suggestions that the Conservatives' goal of reducing net migration below 100,000 had contributed to the Windrush problem.
She said she never agreed to use removal targets for migrants, adding that the regional targets used by the Home Office "were not published targets against which performance was assessed".
Ms Rudd said she was not familiar with the suggestion that regional targets were in place.
The once hotly-tipped candidate to become the next Tory leader now seems to be running around like a chaotic hotel manager not dissimilar to Basil Fawlty, trying to plug holes, self-inflicted from the incompetent running of her Home Office department.
Quoting Theresa May, when in opposition had said she was "sick and exhausted of ministers not taking responsibility", Lammy's colleague Paula Sherriff asked if Rudd would "do the decent thing and resign?"
The PM's official spokesman said the idea of governments setting removal targets was something that had happened for "a number of decades".
Asked if Ms May supported the principle of targets, he said: "Dealing with issues of illegal immigration and taking enforcement action has been a priority for a succession of governments".