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.
Speaking at a press conference to mark the closing of this week's Commonwealth heads of government (CHOGM) summit, the prime minister said compensation would be offered to resolve the "anxieties and problems" of those who had been threatened with deportation despite having lived in the United Kingdom for decades and qualifying as British citizens.
Video: Who are the Windrush generation?
"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", she said.
Ms Cooper said they would "pursue the evidence wherever it goes", raising the prospect of Mrs May potentially being summoned to give evidence.
Sonia Williams is one of the first of the Windrush immigrants to formally be recognised as a British citizen after the Home Office scandal that left hundreds of people facing deportation.
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They have lived here for several decades, but have fallen foul of changes introduced to the immigration system while Theresa May was home secretary, which aimed to make it easier to identify and expel illegal immigrants from the UK.
Her comments came as details emerged of long-standing United Kingdom residents who said they had been refused re-entry to Britain after visits to the Caribbean.
It has been suggested by some that compensation should be paid to those caught up in the row.
There has been mounting pressure on May for her tightening of immigration law - to create what she dubbed a "hostile environment" on the issue - when she was running the Home Office.
The comments from Ms Cooper came after a former head of the civil service claimed the PM's pursuit of a "hostile environment" for illegal immigrants in Britain was seen in Whitehall as "almost reminiscent of Nazi Germany".
Mr Gove, who held a number of Cabinet positions during the period, said he did not "recollect any such conversation of the kind that he refers to during that time". Sky News would like to hear from you.