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.
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.
Recent changes to immigration law in the UK, which requires people to have documentation to work, rent a property or access benefits, has highlighted the issue and left people fearful about their status.
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."
My parents arrived in this country from Guyana and I stand in Parliament as a proud son of the Windrush. Thank you to 140 colleagues including @jeremycorbyn @HackneyAbbott @AngelaRayner @YvetteCooperMP @ChukaUmunna @DawnButlerBrent for joining me in writing to @theresa_may today https://t.co/O5Wl6TeGbX— David Lammy (@DavidLammy) April 16, 2018