But Downing Street stressed that, while Mrs May was aware of targets for illegal immigrant removals during her time as home secretary, she ceased receiving this operational information when she left the Home Office.
Britain's new Home Secretary, Sajid Javid, has pledged to do all he can to help victims of the Windrush scandal - mostly Caribbean immigrants and their descendants denied basic rights after being falsely branded illegal.
In an attempt to derail the Labour motion, May had used prime minister's questions earlier in the day to say home secretary, Sajid Javid, would be "announcing a package of measures to bring transparency on the issue".
May declined to answer directly when she was asked whether she should have taken personal responsibility for the scandal instead of Rudd.
May and Rudd apologised for the treatment of the Windrush generation during the recent Commonwealth summit attended by leaders from Caribbean countries, among others.
"Once the Prime Minister ceased to be home secretary, she would not have received any operational detail of that kind", the Prime Minister's spokesman said. "I don't like the phrase "hostile'".
"The government was warned that the negative outcomes for Commonwealth citizens would be a effect of the hostile environment policy", concluded Abbott.
"If you look at what we're doing as a government, and have been doing over the years as a government, what we are doing is responding to the need that people see for a government to deal with illegal immigration", she said.
Javid is the first-ever British Muslim and British Pakistani to get the post - the most important cabinet position after the prime minister's.
Rudd, under growing pressure over the so-called Windrush scandal, told lawmakers last week that there were no targets for the removal of people deemed to be in the country illegally.
The group referred to as the "Windrush generation" relates to a ship named Windrush, which brought hundreds of Jamaican workers to United Kingdom shores in 1948.
She telephoned May to tell her of her decision, but in a letter that officially marked it she wrote: "It is with great regret that I am resigning as home secretary".
The prime minister's critics note that it was May - not Rudd - who was responsible for the changes in immigration rules that led to doctors, landlords and employers checking people's immigration status.
Invited to Britain after World War II, they were given a legal right to remain by a 1971 law.
Rudd was the fifth person to quit the cabinet since the June 2017 snap general election, called by May but which cost the Conservatives their majority in parliament.
The issue centres around thousands of UK-based Jamaicans facing forced deportations due to lack of documentary evidence that they had the right to live and work in Britain because they arrived pre-1973, when stricter new visa norms came into force for all Commonwealth nationals migrating to the UK.