The Prime Minister confirmed the practice was in place when she was home secretary, but Downing Street said Mrs May had no information on targets for immigration removals when Ms Rudd wrongly told MPs last Wednesday that they didn't exist.
"This never should have been the case and I will do whatever it takes to put it right", he said.
Outrage at their treatment has piled pressure on May, who was home secretary between 2010 and 2016 and introduced tough immigration policies meant to make Britain a "hostile environment" for unauthorized migrants.
She said: "It's wrong to think the 100,000 net migration target is the problem here".
Earlier, Mrs May defended the existence of targets during a local election campaign visit to Greater Manchester: "When I was Home Secretary, yes, there were targets in terms of removing people from the country who were here illegally".
Sajid Javid is the first lawmaker from the black, Asian and minority ethnic community in Britain to be appointed as Home Secretary.
But ironically, and perhaps unfairly, for Rudd, leaked documents also indicate that the policy of forced removal of illegal immigrants was aimed at recent arrivals, mainly from India and Pakistan, and not for the Windrush generation, though some were affected by it.
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Luigi Di Biagio took charge of the team for their friendlies against Argentina and England in March on an interim basis. With Inter he won three Serie A titles and he guided City to their first Premier League triumph back in 2012.
James Brokenshire - who stepped back from his role as Northern Ireland secretary while he received treatment for cancer - will replace Javid at the Ministry for Housing, Communities and Local Government, while global development secretary Penny Mordaunt will take on the women and equalities brief vacated by Rudd.
He replaced Amber Rudd, who resigned on Sunday night following a crisis over the threatened deportation of members of the Windrush generation, whose families moved to Britain from the Caribbean in the 1950s and 1960s. Both Rudd and Prime Minister Theresa May have been heavily criticized for their "hostile environment" policy toward immigrants.
Javid already sought to distance himself from the Home Office fiasco in a highly-personal Sunday Telegraph interview published over the weekend.
He said his parents, who migrated from Pakistan to the United Kingdom, would be incredibly proud of his appointment, before adding: "But I haven't called my mum yet".
But Rudd now might join forces with other pro-EU Conservative lawmakers, further reducing May's strength in parliament, where she lost her party's majority at an ill-judged election past year. "We must urgently right this historic wrong".
Javid's appointment provides the Tory party with an opportunity for a reset on immigration - an issue which has dogged the government.