The cabinet has yet to discuss any proposals to curb immigration after Brexit despite the looming start of talks with the EU, the Home Secretary has admitted.

The approach was criticised as “baffling” by the head of a Commons committee, who accused the government of failing to be “honest and open” with the public on one of its key concerns.

The negotiations on Britain’s long-term relationship with the EU are poised to include immigration, with Brussels’ draft proposals seeking an “ambitious” agreement.

But Amber Rudd, giving evidence to the Home Affairs Committee, said the UK’s proposals would not be published until the end of this year – as the Brexit negotiations are due to finish.

And, asked if there was a “cabinet position” on future immigration – or even on whether it should be included in the future partnership talks – she replied: “It is for the Prime Minister to decide that.”

Yvette Cooper, the committee’s chair, warned that a failure to negotiate on immigration would leave the UK facing an inferior, “Canada-style” deal with the EU.

And she raised fears that MPs would be denied a say, warning: “How will parliament ever see what those objectives are? It is slightly baffling that there isn’t more of a plan as to what should happen on immigration.”

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Ms Rudd also came under pressure over an apparent backtracking on the government’s pledge to stop EU citizens being rejected for permanent residence, because they lack comprehensive sickness insurance.

There were protests that EU nationals were unaware of the requirement – prompting the announcement, last year, that it would be dropped as part of a “streamlined” process.

Stuart McDonald, an SNP MP, demanded to know why the promise not to require sickness insurance was missing from the draft Brexit transition deal if it had been “waived”.

The Home Secretary said it was missing because the government was “confident that EU countries will do the same” – but said she would ask the Health Secretary for clarification.

Ms Rudd also said the government would not be intervening on behalf of the 1.2 million UK nationals in EU countries who have warned they are “in the dark” over their future rights.

She said it would be “up to individual countries” to decide their freedom of movement rights. The chief concern is whether the right to live and work in another EU country will end.

The future partnership talks are about to start, with the EU determined to reach an outline agreement by the autumn – to allow ratification by the European Parliament by Brexit Day next March.

But Ms Rudd confirmed a white paper would not be issued until the end of 2018, with a bill not expected until next year – just weeks before Britain leaves the EU.

She claimed the urgency had “to a certain extent been removed” because the rights of EU citizens in the UK would be “protected” by the withdrawal agreement.

Mr McDonald suggested the delay meant it was inevitable that free movement of EU citizens would have to continue beyond the end of the transition period, intended to be at the close of 2020.

Similarly, on Tuesday, Theresa May accepted the UK might not “be able to achieve” a new customs deal by the time the transition runs out.