People who backed Brexit in the belief it would lead to a cut in immigration into the UK were voting for something that is in effect impossible, a former member of David Cameron’s cabinet has said.

Stephen Crabb, a former work and pension secretary, said that Theresa May urgently needed to outline a new set of values for a post-Brexit immigration system, or the public may face a rude awakening.

“For many, a vote for Brexit was indeed a vote to take back control and return to Westminster the full tools to cut immigration,” Crabb said in a article for the Guardian.

“The problem is that, set against the popular expectation that Brexit means cutting immigration, there is nothing on the horizon to suggest that achieving any significant reduction is achievable or even desirable.”

Crabb argued the fallout from US president Donald Trump’s “toxic immigration decree” had increased the need for a clearer definition of British values towards immigration.

Crabb’s intervention comes after May published a white paper setting out her approach to Brexit. It contained little detail about the UK’s future immigration policy but made clear there will need to be legislation to form a new system – and that any changed approach would be phased in.

Net migration has been rising in recent years, despite Cameron’s stated policy of reducing it to the tens of thousands. It was at a near-record high of 335,000 in the 12 months to the end of June, the most recent figures available.

Crabb also joined a number of other Tory MPs on both sides of the Brexit debate who are calling for the UK to guarantee the rights of EU nationals – an issue that could be the subject of a House of Commons rebellion during the passage of the Brexit bill next week.

“By recognising their value to our economy and society, and the sheer inhumanity and impracticality of ever thinking that these families and individuals could be required to leave the place they call home, we can take the issue off the table altogether. Now that would be a powerful and positive statement of our Brexit values,” he said.

He said the economy is continuing to absorb new labour arriving in the UK and no minister has been able to point to any group of foreign workers who should or would not be in the country after Brexit.

On top of that, the governments of Australia, New Zealand, Canada and India have already indicated that they will be looking for easier access for their workers to come to the UK as part of any deal, Crabb added.

“There could be another rude awakening for the public when they realise that Brexit will not mean a cut in immigration after all,” he said. “It would be far better for the government to be upfront with the British public now and begin explaining current labour market and demographic realities. The previous pledge to cut immigration to the tens of thousands is, in truth, increasingly irrelevant.”

He suggested the UK should take a more sophisticated approach than a net migration target, perhaps starting with taking overseas students out of the numbers.

May has been clear that cutting immigration is a priority for the government that will require the UK to abandon free movement rules within the EU and leave the single market.

The prime minister has stuck to Cameron’s target of reducing net migration to the tens of thousands but has not put a timeframe on when this will happen.