Yvette Cooper is open to a “one in one out” youth mobility scheme with the European Union, according to reports. The Home Secretary had been seen as the biggest opponent over fears it would push up migration figures.

But government sources told The Times that Ms Cooper was supportive of the plan as long as it was "capped" to ensure that there could be no return to pre-Brexit freedom of movement. One option would be to limit numbers so young Europeans coming to live and work in the UK did not exceed British people going to Europe, while another would be to set an annual cap.

The EU has pushed for a youth mobility scheme ahead of a UK-EU summit planned for May 19 as Sir Keir Starmer pursues a Brexit "reset". However the move is likely to spark a backlash from Brexiteers.

More than 60 Labour backbenchers signed a letter calling for "a new and bespoke youth visa scheme" for UK and European citizens under 30. But Cabinet Office minister Nick Thomas-Symonds yesterday insisted the move was not in the Government's plans.

Mr Thomas-Symonds told the Commons in response to a question from the Lib Dems: "On the issue of a youth mobility scheme, it is not part of our plans. We have always said we listen to sensible EU proposals, but we will not go back to freedom of movement.

"Where I do agree with her, is about concrete proposals and concrete progress on the 19th of May, and we are looking to secure that new partnership with the EU that will make our country safer, more secure and more prosperous."

A Home Office spokesman said: “These are matters for the Cabinet Office. No proposals have been put to the home secretary or suggested by her.”