Any impression of an overstay clampdown must be dispelled, May said, following reports that Caribbean migrants invited to fill labor gaps in Britain after World War Two were later ostracized, some due to changes in the law during May's six-year term as interior minister.
Some of the invited migrants had reportedly been denied British passports and health treatment despite having paid taxes for decades.
Read more: May's 2017 election campaign, migration curbs
Britain 'has heard us'Arriving in Downing Street, Barbados ambassador Guy Hewitt said: "We are now in discussion mode. The [British] government has heard us."
Jamaican Prime Minister Andrew Holness said he looked forward to a "speedy implementation" of the British government's proposed solution.
Lacking documentsAfter World War II, Britain encouraged Caribbean citizens to come to the country to fill labor gaps and aid the rebuilding efforts. At the time, they were considered citizens of the British Empire.
The first ship to arrive was the "Empire Windrush" in 1948, hence the phrase Windrush Generation. In 1971, with most of the Empire disbanded, laws were changed to make it more difficult for citizens of Commonwealth countries to move to the UK. But many of the Windrush Generation remained in the UK, without British passports and with now-defunct documents from their home countries.
Some of them were then later treated as illegal migrants, especially in the aftermath of reforms during May's tenure as interior minister designed to crack down on illegal immigration. The Guardian newspaper said a former House of Commons cook Paulette Wilson, who arrived in Britain aged 10, was sent to a migration detention center last year.
Read more: Brexit Diaries 33
Almost half a million people left their homes in the Caribbean to live in Britain between 1948 and 1970, according to estimates from Britain's National Archives.
Online petitionAs May met the Caribbean leaders on Tuesday, calls for an amnesty had already drawn 136,000 signatories to an online petition and more than 140 parliamentarians had signed a similar letter.
On Monday, British interior minister Amber Rudd apologized and junior home minister Caroline Nokes admitted some might have been deported. Rudd promised to sort out their paperwork for free.
"Frankly, some of the ways they have been treated has been wrong, has been appalling and I am sorry," Rudd said.
On Tuesday May's spokesman claimed no deportations had taken place once the error was raised, adding that Britain had already voiced regret in a letter to the 12 nations.
Opposition Labour Party leader Jeremy Corbyn said: "It's disgraceful that the rights of the Windrush Generation have been brought into question by this government and that some have been wrongly deported."
Meager share of UK trade with CommonwealthOn Monday, British Trade Minister Liam Fox acknowledged that the United Kingdom needed to "re-invigorate" its partnerships within the 53-nation Commonwealth as it prepares for its exit from the European Union next year.
Overall, the Commonwealth, even with Canada and India up front, accounts for only a 10th of Britain's trade. In contrast, Britain's trade with the EU via its single market makes up more than 40 percent of British exports.
The Commonwealth links 2.4 billion people over five continental regions, from Tonga and Vanuatu in the Pacific to wealthy Australia and vast India.
ipj/msh (Reuters, AP, AFP)