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 row erupted after it emerged relatives of migrants from Commonwealth Caribbean countries who settled in the UK from the late 1940s to the 1970s had been declared illegal immigrants if they could not provide a range of documentation proving they had lived in the UK continuously.
Some have been threatened with deportation, lost their jobs or been refused access to medical treatment.
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."
Who knew what, when?
BBC political editor Laura Kuenssberg
As the Windrush scandal has unfolded it has been said again and again that the Home Office was not aware of the problem as a pattern where one particular group was being scooped up into the immigration system unfairly, on multiple occasions.
Yes, individual cases reared their heads, we know they were raised as far back as 2011. But Amber Rudd repeated to MPs again today she wasn't aware of the issue as being "systemic", even though she "bitterly regrets" not spotting the pattern.
Ministers' defence has been: we simply didn't realise, but we know now it was a big mistake.
That defence looks a bit shakier tonight. It's not just that some of the government's own work on immigration policy warned in advance of the rules changing that it could hypothetically hit the wrong people.
The Home Office is investigating whether anyone has been wrongly deported. Ms Rudd told the committee that so far 7,000 out of about 8,000 records dating back to 2002 had been checked with no wrongful removals discovered so far.
And she rejected suggestions that the Conservatives' goal of reducing net migration below 100,000 had contributed to the problem.
"It's wrong to think the net migration target is the problem here," she said.
"The problem here is that people were not properly documented."
Earlier 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.