The transition period could last between two and four years after Britain leaves the European Union as expected in 2019, the Times and the Guardian reported, citing anonymous sources.
"If you ask business when they want to see it agreed, they'd say tomorrow," a senior cabinet source told the Guardian. The Times quoted "a British source close to the negotiations" with Brussels.
Curbing EU immigration was a key argument for the "Leave" campaign in last year's referendum in which Britain voted to end four decades of EU membership.
Around 250,000 EU nationals move to Britain every year -- mainly from eastern and southern Europe -- and a total of around 3.2 million live in the country.
May's government has been riven by infighting between supporters of a clean cut with the European Union and those who want a "soft" Brexit that would retain much stronger European trading ties.
Granting free movement of people could allow Britain greater access to the European single market even after it has formally left the EU.
CBI leader Carolyn Fairbairn was among business leaders who met with May at her Downing Street office yesterday following complaints from many firms about a lack of clarity in government plans.
May "reiterated that the government's overarching goal is for a smooth, orderly exit culminating in a comprehensive free trade deal with the EU, with a period of implementation in order to avoid any cliff-edges," a Downing Street spokesman said.
Francis Martin, head of the British Chambers of Commerce, who also took part in the talks, said a transition period was a priority for business.
"The prospect of multiple, costly, adjustments to trading conditions is a concern for many, so starting discussions on transition arrangements as soon as possible would go a long way to boost business confidence," he said.
The EU's Brexit negotiator urged Britain yesterday to provide more clarity on key issues after the second round of talks wrapped up in Brussels with "fundamental" differences remaining.
Michel Barnier said after talks with his counterpart David Davis that the two sides were still at odds over Britain's divorce bill and over the rights of European citizens living in Britain.
The next round of talks is expected to start on August 28.
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