MPs will be able to force Theresa May to accept a fresh referendum on Brexit in a showdown vote as early as the autumn, a minister has conceded.

In a surprise admission, Steve Baker said the crucial vote on the exit deal would not – as expected – be a “take-it-or-leave-it” choice, because “parliament can always seek to amend motions”.

The Brexit minister agreed a possible amendment would be for parliament to only approve the withdrawal agreement struck with the EU “subject to a second referendum”.

It means MPs will have the chance to send the controversy back to voters before parliament has given its initial support and - crucially - with plenty of time before Britain leaves the EU, in March next year.

The revelation delighted supporters of a further referendum, who launched a high-profile cross-party campaign for the vote only days ago.

“Theresa May has tried to pretend that any vote in parliament on a Brexit deal would be a take-it-or-leave it proposition,” said Stephen Doughty, a Labour MP and People’s Vote campaign supporter.

“Now her own ministers have admitted that’s not true. Parliament can, if it chooses, let the people have their say on the deal, and with public support for a People’s Vote growing by the day, that’s exactly what parliament should do.”

Caroline Lucas, the co-leader of the Green Party and a backer of the anti-Brexit Best for Britain campaign, said that “many MPs from across the political divide” would wish to amend the motion.

“It’s now down to parliament to flex its muscles on this issue, and give people a say on this monumentally important decision,” she said.

Mr Baker’s admission came in evidence to a Lords committee investigating the nature of the so-called “meaningful vote” on the final deal, which will take place later this year, or early in 2019.

The prime minister, and other ministers, have repeatedly insisted that while MPs can reject the agreement reached with the EU, that would simply mean the UK crashing out with no deal at all.

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But Mr Baker, when asked if the motion “would be amendable”, told peers: “I believe the position is that parliament can always seek to amend motions.”

The minister was then asked specifically what would happen if an amendment requiring a fresh referendum was carried, replying: “That would be an extremely controversial situation, of course.”

Mr Baker acknowledged such an amendment could not be ignored, adding: “We would be in the position where we would have to look extremely carefully at what the amendment precisely was.”

And he held open the possibility of attempts to change the nature of the deal – which, if successful, would send the prime minister back to the negotiating table, it has been argued.

“Clearly, the range of amendments that is possible is within the bounds of the imagination of all members of parliament,” the minister said.

Speaking to the Lords constitution committee on Wednesday, Mr Baker suggested parliament had a duty to respect the referendum result and not to “frustrate that process”.

“We will leave, there will not be attempts to stay in by the back door, there will not be an attempt to reverse the result,” he insisted – but he admitted it was “a political point, rather than a constitutional point”.

Ms May has firmly rejected a further referendum, but some pro-EU Tories believe she could yet accept one if it appears the only way to keep her warring party together on Europe.

Similarly, although Jeremy Corbyn has said Labour does not back another referendum, he has left the door open to a change of mind.

There is also evidence that support is growing for a referendum on the Brexit deal, amid continuing confusion about both the planned transition period and the final agreement with the EU.

A recent poll for Best for Britain found that 44 per cent of people want a vote – a clear eight points ahead of the 36 per cent who reject a further referendum.

Opinion appears to be shifting as the negotiations remain bogged down on how to avoid a hard border in Ireland and with the details of a future trade deal unlikely to even be discussed until after departure day.

Strikingly, last month David Davis predicted that parliament would vote down the deal unless the trade terms were broadly agreed – but the EU has insisted there is no time before its autumn deadline.

The constitution committee was told the vote would take place “as soon as possible” after an expected autumn agreement in Brussels, and that it should be supported “in the national interest”.

If parliament backs that motion, it will be followed by a Withdrawal Agreement and Implementation Bill (Waib) to put the agreement into legal effect.