Some “terrible mistakes” were made in cases involving the Windrush generation facing deportation from the UK, says immigration minister Caroline Nokes.
Many long-term immigrants who arrived from the Commonwealth as children have been told they are here illegally.
The BBC understands Home Secretary Amber Rudd plans to set up a team in the Home Office to help those affected.
It follows a reversal by the prime minister, who will now discuss the issue with other Commonwealth leaders.
A meeting of leaders, which will take place this week, was announced amid growing calls for Theresa May to take action, including a letter from a cross-party group of 140 MPs.
Labour MP David Lammy tweeted that the meeting was a “small U-turn”, adding that he wanted the government to “guarantee the status of all the Windrush children caught up in this crisis” by the end of the day.
Labour leader Jeremy Corbyn tweeted it was “disgraceful” that the rights of the Windrush generation had been brought into question, calling on Mrs May to “answer serious questions about how this happened on her watch”.
Mrs May’s spokesman said the prime minister was clear that “no-one with the right to be here will be made to leave”.
He added that the PM is “aware that many people are unlikely to have documents that are over 40 years old”.
London mayor Sadiq Khan said he welcomed Mrs May’s decision to meet with other leaders, but added: “She must now go further and make an immediate commitment to recognise and secure the rights of Commonwealth citizens.”
Thousands of people arrived in the UK as children in the first wave of Commonwealth immigration 70 years ago.
They are known as the Windrush generation – a reference to the ship, the Empire Windrush, which brought workers from the West Indies to Britain in 1948.
Under the 1971 Immigration Act, all Commonwealth citizens already living in the UK were given indefinite leave to remain – but the right to free movement between Commonwealth nations was ended from that date onwards.
However, the Home Office did not keep a record of those granted leave to remain or issue any paperwork confirming it, meaning it is difficult for the individuals to now prove they are in the UK legally.
The BBC understands that Home Secretary Ms Rudd will make a statement to the House of Commons on Monday afternoon to confirm the creation of a new team in her department to help the Windrush generation and ensure no-one loses their access to public services and entitlements.
She is also expected to waive fees so that those affected will not have to pay money for new documents to prove their status.
Mr Lammy has also tweeted that he has secured an urgent question in the Commons on Monday to push Ms Rudd for answers.
Speaking to BBC Radio 4’s World at One, Ms Nokes said the Windrush generation had “contributed an enormous amount to our community [and] to our society” and that the government had “an absolute responsibility to make sure there are no more of these mistakes”.
Asked by ITV News if any people had been deported as a result of these “mistakes”, Ms Nokes said: “There have been some horrendous situations, which as a minister have appalled me.”
Told by the reporter “that’s a yes” and asked how many, she said: “No, I don’t know the numbers, but what I’m determined to do going forward is we’ll have no more of this.”
Penny Mordaunt, the international development secretary, said she wanted to reassure those affected, telling BBC Radio 4’s Today programme: “People who are in that situation, there is absolutely no question of their right to remain, and their right to gain access to services such as healthcare.”
A letter to the prime minister, co-ordinated by Mr Lammy, called for a “swift resolution of this growing crisis”.
It said: “We urge you to guarantee the status of all Commonwealth nationals whose right to remain is protected by law and to provide an effective, humane route to the clarification of their status.
“What is going on is grotesque, immoral and inhumane,” he said.
It was signed by 140 MPs including Labour leader Jeremy Corbyn and Conservative MP Sarah Wollaston.
Omar Khan, from the Runnymede Trust charity which has been involved in trying to tackle this issue, said the onus should be on the Home Office to help people find the documents they need.
He also called for an extension of legal aid to these cases.
He told the BBC’s Victoria Derbyshire programme: “These are individuals who do have legal rights – this is not really an amnesty. The issue is their ability to prove it through documentation is now quite difficult.”
Guy Hewitt, Barbados high commissioner, told the BBC: “I have held as a great honour the fact that I am the first London-born high commissioner for Barbados.
“This is the first time I have felt that the country of my birth is saying to people of my region ‘you are no longer welcome’.”
The Migration Observatory at Oxford University estimates there are 500,000 people resident in the UK who were born in a Commonwealth country and arrived before 1971.
People born in Jamaica and other Caribbean countries are thought to be more affected than those from other Commonwealth nations, as they were more likely to arrive on their parent’s passports without their own ID documents.
Docks with 482 Jamaicans on board
Many have never applied for a passport in their own name or had their immigration status formalised, as they regarded themselves as British.