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The Home Office and Downing Street were told in 2016 about problems faced by the Windrush generation, the BBC understands.
They were alerted after the Barbados government raised concerns with the Foreign Office, BBC political editor Laura Kuenssberg said.
Labour has urged Home Secretary Amber Rudd to quit over the Windrush saga.
Ms Rudd told MPs she "bitterly" regretted not recognising the "systemic" problem sooner.
The Windrush cases - which include anyone who moved to the UK from a Commonwealth country before 1973 - are in the UK legally but some have been threatened with deportation if they did not have paperwork to prove it.
Their plight has provoked a storm of criticism for the government, with Prime Minister Theresa May apologising for their treatment.
In April 2016, the then Foreign Secretary Philip Hammond - who is now the chancellor - was told by Caribbean ministers about immigrants facing deportation despite having lived in the UK for most of their lives, and the BBC understands a report about their concerns was passed to the Home Office, which was led at the time by Mrs May.
It is not clear at what level the concerns were raised.
The government has set up a task force to help those affected formalise their status.
So far 3,800 calls have been made to the helpline, of which 1,364 were potentially Windrush cases, MPs were told on Wednesday.
Ms Rudd told the Commons Home Affairs Committee that she had known about the problem for months and said officials were still checking how many people had been detained over their supposed immigration status.
"I look back with hindsight and I'm surprised I did not see the shape of it sooner," she said. "I bitterly, deeply regret that I didn't see it as more than individual cases that had gone wrong that needed addressing. I didn't see it as a systemic issue until very recently."
The issue dominated Prime Minister's Questions in the Commons, with Labour leader Jeremy Corbyn urging Mrs May to abandon the government's "cruel" immigration policy and abandon "bogus" targets.
Mrs May rejected claims she was "ignoring" the plight of the families of Caribbean migrants who have had their residency rights questioned, repeating the government's promise they and others from Commonwealth nations who came between 1948 and 1973 would now be offered British citizenship free of charge and would be helped in clarifying their status.
But she said a distinction should be drawn between those people who had settled in the UK legally and contributed to British life and those with no right to be in the UK.