Amber Rudd's ex-deputy says he often spoke to the home secretary about work being done to increase migrant removals but not "detailed numbers or targets".
There was speculation that Ms Rudd, who is now fighting for her political life, was on the brink of resignation after she took almost eight hours to respond to a Guardian report on the leaked document.
Her handling of the controversy has been heavily criticised by MPs, with over 200 of them coming together to sign a letter accusing her of making up immigration policy in an effort to overcome the scandal.
May's government recently said the Home Office destroyed landing documents for Caribbean immigrants, meaning some of them and their children faced deportation after decades living in Britain because they don't have the correct paperwork.
What is the Windrush scandal and how did the fiasco develop?
Around 500,000 people, now residing in the United Kingdom arrived there before 1971 and originate from Commonwealth countries, according to Oxford University's Migration Observatory.
1973 - A new immigration Act comes into force putting the onus on individuals to prove they have previously been resident in the UK.
"Amber is a great home secretary", he said.
The prime minister has another problem, too.
At a meeting on 17 April, May told representatives of the 12 Caribbean members of the Commonwealth that she took the treatment of the Windrush generation "very seriously".
"When Amber Rudd says she didn't see something, I know she didn't see it".
Mr Lewis, who is now the Conservative Party chairman, said that while they had had weekly discussions about immigration, they never mentioned targets.
Mr Gove, appearing on BBC Radio 4's Today programme, also backed Ms Rudd, saying she was "a highly talented and highly effective minister".
Communities Secretary Sajid Javid, whose parents emigrated from Pakistan in the 1960s, said the Government was committed to "put things right".
It had emerged in the past 24 hours that she may have misled a committee of MPs over whether her department had targets for removing a certain amount of illegal immigrants from the country over a given period of time.
'She was aware of the increase that we wanted to remove nationally, those internal targets were not something she was aware of.
When she did respond, in four highly defensive tweets, she apologised once again for the Windrush scandal and said she should have been known about the targets.
Diane Abbott said the home secretary may have breached the ministerial code by not giving accurate and truthful information to Parliament.
Amber Rudd has resigned as home secretary, Downing Street has announced.
"Brandon Lewis's attempt to save the Home Secretary's career by hiding behind semantics is insulting for the people who have been affected by the Tories' heartless policies", she said.
'Beneath the spin, he let the truth slip and sealed her fate.
"The Home Office was fully aware that lives were being ruined as a result of their "hostile environment" but pressed ahead anyway".
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