Yvette Cooper, the committee’s chair, said: “Immigration was one of the central issues during the referendum and it divided the country, but sadly there has been no attempt by the government to hold any kind of sensible debate on it or build any kind of consensus on immigration since.
“The misinformation and tensions over immigration during the referendum campaign were deeply damaging and divisive.
“It is essential that does not happen again, and those who exploited concerns over immigration during the referendum need to be more honest and more responsible when it is debated in the run-up to the final deal.”
The report comes as the number of people supporting The Independent’s campaign for a Final Say on any Brexit deal nears 400,000 after just six days.
In the 2016 campaign, Vote Leave was accused of Ukip-style rhetoric when it claimed Turkey and other countries were set to join the EU by 2020, allowing millions of people to move to Britain under freedom of movement rules.
It also released a controversial video of David Cameron, the then prime minister, advocating Turkish membership of the EU, interspersed with scenes of fighting in the country’s parliament.
But, despite immigration dominating the campaign – and the issue’s likely inclusion in the negotiations on Britain’s long term trading arrangements with the EU – no proposals have followed.
Ms May has repeatedly insisted free movement will end, but knows that moving beyond that slogan is certain to alienate Tories on one side of the Brexit divide.
The white paper is not due until the end of the year, with an immigration bill not expected until next year – just weeks before Brexit day, on 29 March.
In today’s report, the home affairs committee also:
* Says EU migration will “remain important in future”, because the UK’s need for skilled workers and its “shared economic, social and cultural bonds” with the EU.
* Urges the government to “explore” immigration curbs that are possible even with membership of the EU single market – including an “emergency brake” if a country is in “difficulties”.
* Concludes some curbs – including to prevent undercutting of wages and on EU citizens without work – could be introduced immediately.
* Repeats its call for the government to axe its “tens of thousands” migration target, rather than make it an objective of post-Brexit migration rules.
* Urges ministers to consider a regional immigration system, to give local areas “flexibility in their approach”.
“We found there were a much wider range of possible precedents and options for immigration reform than people often talk about – including options that could be combined with participation in the single market – that we believe the government should be exploring further now,” Ms Cooper added.
The delay to immigration proposals has already prompted alarm among pro-Brexit MPs that Ms May is planning another sellout, this time over freedom of movement.
Some members of the government believe conceding ground may be the only way to secure an advantageous trade deal – perhaps by offering EU citizens preferential access to the UK.
Critics believe this would amount to “freedom of movement in all but name”, calling instead for EU citizens to fall under a single international visa regime.
Last year, the prime minister axed the Vote Leave proposal for an “Australian points-based system”, because it does not set a limit for reducing immigration but is instead based on a country’s economic need.
The white paper that followed the Chequers deal said only that a plan would follow a September report by the independent Migration Advisory Committee (MAC), which would “provide important evidence”.
Ministers agreed last year that free movement of EU citizens would have to continue until the end of the post-Brexit transition period, intended to be at the close of 2020.