If it’s not Jacob Rees-Mogg, it’s Jean-Claude Juncker, the double-barrelled duo giving Theresa May both barrels within 24 hours of each other. No sooner had the monocled mutineer called for May to resign after she had seen off his motion of no confidence than the head of the EU commission was accusing the prime minister of being “nebulous”, as Europe’s heads of government refused to give her even a few crumbs of political comfort to take back home for Christmas. Rebuffed in both Westminster and Brussels, May is coming to resemble a figure from a medieval morality play, scorned and rejected wherever she turns.

May ploughs on but referendum may soon be her least bad option Read more

The result is that a bleak clarity is emerging from the Brexit fog. There is no majority in the Commons for May’s deal. And yet the deal will not be substantively improved by the EU. So Brexit is stuck, parliament’s minimum falling short of Brussels’ maximum. No deal is possible, which surely makes “no deal” possible.

Except that MPs and ministers alike insist they will not allow the country to fall off a cliff, crashing out with no deal on 29 March. They have heard the words of the former head of the World Trade Organization, Pascal Lamy, warning Britons that to go from the European single market to trading on WTO terms will be like being demoted from the Premier League to the old fourth division. Given all that, May – or someone else, if she cannot do it – will have to come up with something else.

Which is why the probability of a second referendum, once the quixotic goal of a handful of unreconciled remainers, is increasing with each passing day. May doesn’t need to be persuaded by the democratic logic of seeking popular approval for Brexit as it actually exists, as opposed to the slogan-on-a-bus version the public voted on in 2016. All she needs to decide is that a referendum is her only way out of the current impasse.

It’s not just Brexiteers, fearful of losing the prize they won against the odds two and a half years ago, who dislike that possibility. Plenty of remainers too are nervous about a contest they fear they could lose, thereby closing the door for good on a close relationship with Europe. And others, many of them Labour MPs in pro-leave seats, tremble at the sheer hostility the prospect of a second vote arouses in the people they represent.

This is a subtler emotion than is usually depicted. It’s often described in simple terms of betrayal, as if leave voters will not forgive their Brexit being “stolen” from them, as Liam Fox puts it. But others describe a quieter sentiment, a resentment that MPs – who are paid to adjudicate on tough and complicated issues – are instead dumping into the laps of voters a conundrum that has stumped the country’s supposed leaders. With that in mind, a couple of Westminster’s more thoughtful figures have been trying to work out how a second referendum, if it comes to that, can avoid being a simple rerun of 2016, with all the animus and division such a ballot is bound to stir. I’m not referring here to the determination many remainers have to fight a better, more effective campaign: less accountancy, more poetry; less cerebral, more emotional; less project fear, more project hope; less establishment, more insurgent. Though all those concerns are well founded.

No, what I have in mind is an effort to make the referendum exercise itself both legitimate in the eyes of leavers and substantively different to the first one. On the first count, it will help if the ballot is seen as a move by May, rather than a demand successfully pressed by the People’s Vote campaign. Much as I admire the latter, leave voters are likelier to accept a plebiscite called by a Tory prime minister seeking to make Brexit happen than one effected by remainers bent on halting it.

But it will also help if the two competing propositions on the ballot – and let’s make the hopeful assumption that no parliament would present a no-deal crash-out as if it were a viable option, since that would be criminally irresponsible – are both significantly different from the leave v remain choices of 2016. It’s easy to see how leave would differ this time around. In place of the abstract, wishful idea of leave – which Brexiteers cast as a pain-free cash bonanza and panacea for all Britain’s ills – would be a concrete, detailed plan for leaving: May’s plan, more or less.

What, though, of the other side of the ballot paper? Remain will be weakened if it looks like an appeal to wind the clock back to 22 June 2016, as if the status quo ante were some unimprovable nirvana. It would be accused of not having listened to a word leavers have said about why they wanted out.

That’s why some are talking of a “reformed remain” option, an alternative to leave that does not smack of a complacent desire to pretend Brexit never happened. For those Labour MPs in leave seats, their chief hope is that they can signal to their voters that both they and the EU itself have heard their concerns on one issue especially: immigration.

Play Video 0:38 Jean-Claude Juncker: 'Our British friends need to say what they want' - video

I admit to finding this effort depressing. I’d rather live in a country where free movement is seen as a cherished right, a two-word promise of liberation – offering Britons the ability to live, work, travel, study and fall in love anywhere across a vast, diverse and rich continent. But millions of my fellow citizens don’t see free movement that way. The choice for remainers is to tell those voters they’re wrong, and lose another referendum – or allay their fears and win.

In that context, how much more powerful would the remain case be next time if it could say that a Britain that chose to stay in the EU would have more say on migration? No one is expecting an opt-out on free movement: the EU would never grant such a thing. But when Yvette Cooper’s home affairs select committee probed what measures would still be compatible with the rules of the single market, it found many already used by other EU member states – a whole range of controls that, mysteriously, neither David Cameron nor May ever took up – as well as several possibilities, legal under EU treaty law, that have never been properly explored. (Since the referendum, the EU has, for example, revised its rules to prevent workers’ pay and conditions being undercut by employees posted from abroad.) Of course, there are multiple problems with this approach. For one thing, who exactly would negotiate this improved remain option with the EU? Brussels will only talk to governments; it would not open up a parallel track to a panel of remain-minded dissidents. Such a process would also take time, though that’s true of any move to a referendum: if this is the path May takes, she will have to rescind or extend article 50.

But what it surely comes down to is political will. On Friday Tony Blair suggested Brexit would be as damaging a blow to the European project as it would be to Britain. Accordingly, the EU27 should want to enhance the appeal of remain. Not necessarily, says Charles Grant, the well-plugged-in head of the Centre for European Reform. Sweden, Ireland, Denmark, the Netherlands and perhaps Germany may want to help, Grant says, perhaps adding “some tinsel and coloured lights” to various EU loopholes on migration, so that remain can boast of having reformed free movement. But the more “hardcore federalists” in the commission, and in France, “have mixed feelings about our departure. For decades, we’ve been the pebble in their shoe. We’ve been such a bloody pain” that some will be relieved when we’ve gone.

Even so, if Britons stare into the abyss of Brexit and decide to step back, voting in a second referendum to stay in the EU, that would be a huge boost to the EU and, given Britain’s size as well as its security and intelligence muscle, a major geopolitical prize. It makes sense to urge the EU to do its bit to ensure that remain has the best shot at winning a fresh contest. I understand why Juncker and friends don’t want to help Theresa May leave. But they have an alternative. They can help us stay.

• Jonathan Freedland is a Guardian columnist