"The problem here is that people were not properly documented", she said.
The target for voluntary departures, which happen when an individual or family tells the authorities of their intention to leave the United Kingdom, is contained in a December 2015 report from the Independent Chief Inspector of Borders and Immigration.
The report also said the Home Office had a process for returning families who had no legal right to remain in the United Kingdom, something which had a "single numerical target".
Asked if she had told May to ditch the target, Rudd said: "I have not discussed that with the prime minister".
The Prime Minister appeared in denial about her role while the Opposition leader does not understand public concerns about illegal immigration.
Rudd also rejected the claim the Home Office had targets for how many people should be deported from the country.
However, the report shows that targets for migrant removals did exist at the time and were split between regionally-based enforcement teams.
The Home Secretary on Wednesday said she was not familiar with suggestions that regional targets for removals were in place. On Sunday, the SNP described Rudd's position as "untenable", while Labour's shadow foreign secretary Emily Thornberry told the BBC's Andrew Marr the Windrush debacle was evidence there was "something rotten at the heart of government". "I didn't see it as a systemic issue until very recently".
"I bitterly, deeply regret that I didn't see it as more than individual cases that had gone wrong that needed addressing", she said.
The home secretary admitted that she did not know how numerous Windrush generation had been detained or if there were removal targets, despite an immigration union boss insisting they existed.