It was doubtful before yesterday whether David Cameron would aim, as part of his renegotiation plan, to reduce access to benefits for EU migrants further. In the wake of Party Conference last month, he was not reported to be aiming to do so. And in the wake of his speech yesterday, his intentions are still unclear: “I am open to different ways of dealing with this issue,” he said – a retreat from a commitment to seek specific changes made in a Downing Street statement the day before.

These are a four-year bar on EU migrants qualifying for in work benefits and social housing, and ending the practice of sending child benefit overseas. The Prime Ministers said yesterday that “we now know that, at any one time, around 40 per cent of all recent European Economic Area migrants are supported by the UK benefits system” – and that the figure indicates that access to benefits is a powerful “pull” factor in increasing immigration from other European countries.

The basis for that 40 per cent figure had not been published by the time the speech was made – which prompted FullFact to make a formal complaint – and, since most EU migrants have been in Britain for more than four years, it is not clear to what degree the “pull” factor would be reduced by Cameron’s proposal in any event. And the Government has previously admitted that it keeps no figures on how many EU nationals claim British welfare payments.

Furthermore, the Prime Minister’s scheme may be illegal under EU law, since it discriminates between the citizens of one member country and the rest: Jeremy Heywood, the Cabinet Secretary, has apparently warned Cameron that this may be the case. The proposal is worth probing in this detail because it touches on one of the main voter concerns about our EU membership: that immigration from other EU countries is too high and can’t be controlled.

“Numbers that have increased faster than we in this country wanted at a level that was too much for our communities, for our labour markets. All of this has to change – and it will be at the very heart of my renegotiation strategy for Europe,” the Prime Minister said during his Party Conference speech last year. “The numbers coming from Europe are unsustainable and the rules have to change,” Theresa May during her address to the conference in October.

The two sets of words are consistent, but the two minds may not be at one. That latter speech was trashed by much of the commentariat – itself London-based, liberal-leaning and immigration-sympathetic – as a cynical leadership manouevre and a signal that the Home Secretary was mulling campaigning and voting for EU exit. At the time, I dismissed neither the leadership nor the Brexit angle – but, supporting May’s view, argued that there was more to her speech than either or both.

She has been Home Secretary for over five years, and is now the longest-serving holder of the office for 50 years. During the course of her tenure, she appears to have become increasingly consumed by the level of migration into Britain, particularly that from the EU, which has shipwrecked her aim of reducing its net level. Just as her 2014 conference speech was a serious argument about security, so this year’s was a serious one about immigration – not a clap line-packed crowd-pleaser.

And just as she is increasingly preoccupied by immigration, so Downing Street is increasingly preoccupied by her. ConservativeHome understands that Cameron and George Osborne are now seriously worried that May – who is after all a holder of one of the four great offices of state – will indeed back leaving the EU, perhaps leading the Leave Campaign altogether. I wrote about the possibility on this site last month. One poll has since found her the public’s choice for the role.

When quizzed about the survey and her intentions on Marr, she refused to be drawn. “Cameron doesn’t understand people who act on principle, while Osborne has contempt for people who act on principle,” said one source, who admittedly is not well-disposed to either. But since there is conversation in Downing Street, at the highest level, about what she may do about Europe, there will also be conversation about what to do with her.

The timing of the referendum and its interaction with any reshuffle is complex. But the Prime Minister and the Chancellor will undoubtedly now be brooding about means of getting the Home Secretary out of the Home Office – either by sacking her outright or, less riskily, by moving her elsewhere (assuming that she would accept demotion). Since they would be slated were one of the great offices not to be held by a woman, they will presumably be casting round for a replacement.

It could be Nicky Morgan, but her leadership ambitions will not have endeared her to the Chancellor. It could be Elizabeth Truss who, unlike the Education Secretary, has been co-opted onto Osborne’s Public Expenditure Committee. But the most likely beneficiary is Amber Rudd, the Environment Secretary. She is a signed-up member of Team Osborne. She is also, as one Conservative MP put it to Andrew Gimson, “a Macmillanite pragmatist”, which will appeal to the Prime Minister.

There will be different views in the Home Office about the degree to which benefits really are an immigration “pull”. One, certainly, is that it is overblown, and that wages and employment are much bigger factors. If you’re a Spaniard who can’t get a job, or a Pole on lower pay, Britain’s booming economy has its attractions. During the coming months, May will have to make up her own mind, and then decide what to do next.