Senior Tories have rallied around Amber Rudd, amid criticism for her failure to know specific migrant removal targets.
The home secretary said she had not seen a memo leaked to the Guardian suggesting she knew of the objectives.
Justice Secretary David Gauke said Ms Rudd "made a mistake but didn't knowingly mislead" while Environment Secretary Michael Gove pointed to the vast sum of emails ministers receive.
But shadow home secretary Diane Abbott continued to call for Ms Rudd to quit.
"I am just surprised that she doesn't seem to take the issue seriously enough to offer her resignation," she told Radio 4's Today programme.
Ms Abbott said it was the decision to set a numerical target for removing illegal immigrants that contributed to problems faced by the Windrush generation, where Commonwealth citizens, who came to Britain in the decades after World War Two were wrongly targeted.
"The danger is that very broad target put pressure on Home Office officials to bundle Jamaican grandmothers into detention centres," she said.
Some of the Windrush generation have been threatened with deportation, lost their jobs or been refused access to medical treatment.
Their plight has sparked a storm of criticism for the government, with Prime Minister Theresa May apologising for their treatment.
Ms Rudd faced calls to resign from her position as home secretary after telling MPs earlier this week there had been no targets for migrant removal then later admitting "local" targets existed.
But in a series of late night tweets on Friday, she apologised, saying that although her office had been copied into the document she did not see it herself.
The memo, seen by the Guardian, from June 2017 set out Home Office targets for achieving 12,800 "enforced returns" in 2017-18. It also said targets had been exceeded for "assisted returns".
She is expected to make a statement in the House of Commons on Monday to respond to what she called "legitimate questions" about illegal migration.
Analysis: Rudd's position 'far from solid'
By Jonathan Blake, BBC political correspondent
One by one cabinet ministers have offered support to their colleague Amber Rudd.
After a tense few hours on Friday evening when her position seemed to be uncertain, there is now an effort to rally round the home secretary.
She has given an explanation and an apology, and for MPs in her own party and the prime minister, that would appear to be enough.
To lose Amber Rudd would leave Theresa May further exposed to criticism of her own record at the home office. It would also upset the delicate balance of opinions on Brexit in cabinet with crunch votes on the government's policy looming at Westminster.
But the home secretary's position is far from rock solid. She has promised a statement to parliament on Monday. For Amber Rudd it may feel like a very long weekend.
The chair of the home affairs select committee, Yvette Cooper, has said "serious questions" need to be answered when Ms Rudd returns to the committee to give further evidence.
"We have obviously been given inaccurate information to Parliament twice now," she said.
Asked whether ministers could cite the volume of emails received as an excuse, Ms Cooper said a "huge amount" of documents are received but systems had to be in place.
The environment secretary said that if these memos were not brought to the home secretary's attention that was "regrettable".
Michael Gove told Radio 4's Today: "There are hundreds of documents which are copied in to a secretary of state's office every day and week.
"She was very clear both in her apology and also in the fact that this specific document wasn't placed in her box, wasn't brought to her attention. It wasn't a matter for decision."
Justice Secretary David Gauke told the programme that Ms Rudd is an "excellent" home secretary.
"She has accepted that she made a mistake, she didn't knowingly mislead the House of Commons, but she accepted she was inaccurate in her statements.
"She's coming back to the House of Commons on Monday to correct the record."
Under the Ministerial Code, if any minister "knowingly" misleads Parliament they are expected to resign.
Downing Street has already pledged its support to Ms Rudd, saying the prime minister has "full confidence" in her.
The six-page document, from Immigration Enforcement Agency boss Hugh Ind, states: "IE has set a target of achieving 12,800 enforced returns in 2017-18, aided by the redistribution of resources towards this area.
"This will move us along the path towards the 10% increased performance on enforced returns, which we promised the home secretary earlier this year."
It adds: "We have exceeded our target of assisted returns. We set an internal target of 1,250 of these returns for 2016-17… we delivered 1,581."