Theresa May has said that members of the Windrush generation who have been treated unfairly by the Home Office are to be compensated "where appropriate".
The PM said this will be offered to resolve the "anxieties and problems".
Some people who came to the UK from the Commonwealth decades ago have been threatened with deportation or refused jobs or healthcare.
Meanwhile, it has emerged that the National Archives hold arrival records for thousands of those people.
These records could help some of the Windrush generation who are struggling to prove they are in the UK legally.
It was revealed earlier this week that thousands of landing cards of Windrush migrants had been destroyed in 2010.
However, immigration lawyer Adam Sinfield said the discovery of the records in the National Archives would be of "limited use" to immigrants who might need to prove their arrival in the UK.
"Their biggest problem remains the bureaucracy of having to prove their continuous residency in the late '70s and early '80s," he said.
On Thursday, Chief Secretary to the Treasury, Liz Truss, told BBC Question Time that the government would "look sympathetically" at compensation requests from Windrush immigrants.
Mrs May's first public mention of remuneration came at a news conference concluding the Commonwealth Heads of Government Meeting (Chogm) in London on Friday.
She said: "On Tuesday, I met with Caribbean leaders, where I gave an absolute commitment that the UK government will do whatever it takes – including where appropriate payment of compensation – to resolve the anxieties and problems which some of the Windrush generation have suffered.
"These people are British, they are part of us, they helped to build Britain and we are all the stronger for their contributions."
More details on the plans are expected to be published within the next week.
Labour MP David Lammy, who has spearheaded the campaign for the Windrush generation, said more detail was needed from the government.
He said that "compensation and reparations" should be provided "to anyone who had lost their job, their pension, access to healthcare, or been detained or deported".
Grenada's Prime Minister Keith Mitchell has also called for "serious" compensation in light of the PM's speech, saying it was "highly significant" she had mentioned compensation.
He said: "It's not just, 'I'm sorry.' People lost a lot, people suffered a lot of pain, and they must be given an opportunity to correct this – some serious compensation.
"If not the person, if they've gone, then the families who have suffered too."
- In Pictures: The Windrush generation
A total of 286 people have contacted the Home Office since Monday, when the department set up a helpline to support those who had concerns about their migration status.
So far, eight have been given permanent residence status.
Are you part of the Windrush generation? Will the National Archives locating the arrival records help your case? Please get in touch by emailing firstname.lastname@example.org.
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