Amber Rudd is fighting for her political life after apologising and admitting she should have known the Home Office had set targets for removing illegal immigrants.
Accused of being responsible for "chaos" in the Home Office, the minister said she was sorry for being unaware of the targets and she will make a statement to MPs on Monday.
Downing Street said the Home Secretary has the "full confidence" of the Prime Minister, while Theresa May's de facto deputy, David Lidington, also publicly backed her.
However Labour and the SNP said her position was "untenable" and called on her to resign.
Shadow home secretary Diane Abbot said: "She failed to read crucial documents which meant she wasn't aware of the removal targets that have led to people's lives being ruined.
"Another apology is not enough, she should take responsibility for chaos in the Home Office and resign."
It emerged on Friday Ms Rudd had been copied in to an internal memorandum which described targets set by the Immigration Enforcement (IE) agency for removing people who had no right to be in the UK.
In a series of tweets she later acknowledged that the disclosure had raised "legitimate questions" and said she would go before the Commons on Monday.
Her statement followed accusations that she had misled Parliament after she told the Commons on Thursday that she had never agreed to specific targets for removals.
Ms Rudd said: "I will be making a statement in the House of Commons on Monday in response to legitimate questions that have arisen on targets and illegal migration.
"I wasn't aware of specific removal targets. I accept I should have been and I'm sorry that I wasn't. I didn't see the leaked document, although it was copied to my office as many documents are."
The leak of the memorandum to The Guardian adds to the pressure on Ms Rudd, who has been under fire for her handling of the Windrush scandal which has seen Commonwealth citizens threatened with deportation, even though they were entitled to be in the UK.
Labour says the Home Secretary is being used as a "human shield" to protect Mrs May, Ms Rudd's predecessor.
"Amber Rudd is hanging by a thread to shield the Prime Minister from her responsibilities as the initial architect of this cruel and callous approach to migration, which resulted in the Windrush scandal," Ms Abbot said.
Senior Labour frontbencher Barry Gardiner told BBC 5Live: "Amber Rudd's sole purpose in Government at this moment is as a human shield for Theresa May."
The shadow trade secretary said Ms Rudd had, in his view, "failed to abide by the ministerial code" as it was her duty to be aware of how the Home Office worked.
A Downing Street spokesman said Ms Rudd had "set out her position clearly".
They added: "The PM has full confidence in the Home Secretary, and the hugely important work she is carrying out at the Home Office."
Mr Lidington, the Cabinet Minister, said that running the immigration system was one of the biggest jobs in Government as he voiced support for the Home Secretary.
"@AmberRuddHR has the determination & resourcefulness to see that job done," he tweeted.
According to The Guardian, the secret internal Home Office document referred to the department setting "a target of achieving 12,800 enforced returns in 2017-18" adding "we have exceeded our target of assisted returns".
The six-page memorandum, prepared by Hugh Ind, the director general of the Immigration Enforcement (IE) agency, last June, was said to have been copied to Ms Rudd as well as to Brandon Lewis, the then immigration minister, and a number of senior officials and special advisers.
It went on to refer to the progress that had been made towards achieving a 10% increase in performance on enforced returns "which we promised the Home Secretary earlier this year".
Ms Rudd had initially denied the Home Office had targets for removals when she was questioned on Wednesday by a Commons committee investigating the Windrush fiasco.
However, after it emerged a 2015 inspection report made clear the practice did exist, she told MPs on Thursday that while the immigration arm of the Home Office had used "local targets" they were "not published targets against which performance was assessed".
According to the document seen by The Guardian, dated June 21 2017, the IE agency made 12,503 enforced returns in 2016-17, which was considered a "success".
The memo 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."
The memo also says it "exceeded our target of assisted returns" - which covers cases such as those where an individual has left the country voluntarily on a flight paid for by the British Government.