In December 2015, when David Cameron wanted parliamentary approval to join air strikes against Islamic State (IS) in Syria, he knew that he would be unable to win the vote with the support of Conservative MPs alone. So he called Graham Jones, Labour MP for Hyndburn. Jones is not well known outside Westminster, or even in the political press. Although he was first elected in 2010, he was not part of what one MP is fond of dismissing as the “Goldenballs” generation of Chuka Umunna or Stella Creasy, who were regularly tipped for the top in the pre-Corbyn era. Instead, Jones spent five years toiling in the opposition whips’ office under Ed Miliband. On the economy he inclines to the left, but on foreign policy he is hawkish. That, plus his status as a whip, recommended him to Cameron. The prime minister wanted to know if there were enough Labour MPs who, regardless of Jeremy Corbyn’s anti-war position, would be willing to vote with the government. The answer: there were.

As far as I know, Theresa May has yet to invite Jones to Downing Street for a cup of tea to discuss her Brexit difficulties. But she is facing a problem similar to Cameron’s with Syria: she will need Labour support to offset a Conservative rebellion.

One problem that May has and Cameron did not is that this time the stakes are much higher. Had Cameron been defeated over bombing IS positions, he would have carried on in office. Now, any Labour MP who bails out the Tories on Brexit risks being traduced for killing off the opposition’s chance of an early general election. (In fact, even a government defeat would not be enough to over-rule the Fixed-term Parliaments Act, although Tory MPs are less relaxed about the damage that the DUP could do should it end its confidence and supply arrangement.)

Another problem that May has and Cameron did not is this: there is no equivalent of Graham Jones among Labour’s pro-Europeans. There are well connected and well organised MPs, such as Alison McGovern, and respected veteran Europhiles, such as Pat McFadden, but there is no one individual who can round up and organise rebel votes.

Pro-European Labour MPs represent the deepest potential pool of supporters of a Brexit bailout because they make up around three-quarters of the total number of Labour MPs to have rebelled against the leadership over Europe. When you talk to MPs in this group, they are candid about their fears that rejecting May’s final deal, however appalling it is, will result in a disastrous and unplanned exit. Food shortages and political crisis would follow.

Labour whips are optimistic, however, about their ability to squeeze this group into compliance. They argue that voting against a deal in November or December would leave enough time both to wound the Conservative government and vote again to prevent a no-deal exit in the new year. That was the message of Jeremy Corbyn and Keir Starmer’s speeches to the Parliamentary Labour Party (PLP) on 15 October. Neither speech was rapturously received but nor was there stony silence. There is also a new mood of defiance among pro-European Conservative MPs. This group regards May’s proposals as the “last chance saloon” for the Brexiteers: reject this and a much softer Brexit might follow.

The Tory Europhiles’ argument is this: May’s proposals are the one version of Brexit that has any chance of passing the Commons with only the votes of the Conservatives and the DUP. If the Brexit ultras of the European Research Group (ERG) reject it, Tory Remainers believe the only option to get a deal through the Commons will be to meet the full price of Labour pro-Europeans: continued membership of the single market and customs union. The ERG would like that even less. The Tory pro-Europeans have received a boost from former foreign secretary William Hague. In the Telegraph, he praised May’s tenacity and suggested she adopt liberal Tory MP Nick Boles’s proposal that the UK join the European Free Trade Association (Efta) and take interim membership of the EEA if May’s proposals founder. (Call it a medium-soft Brexit.)

Hague is in close communication with Downing Street about his columns, which further adds to the optimism of pro-European Conservatives. In turn, Labour MPs who want a soft exit will, in the hope of something better emerging, feel emboldened to roll the dice by voting down whatever deal May puts before the Commons.

As for Labour’s small band of Brexiteers – the six sitting Labour MPs who campaigned for a Leave vote, plus Frank Field, now an independent – they are an ideologically heterodox bunch with no leader. And they are usually a reliable supply of extra votes on European issues for the Tories in a tight spot. But a Brexit that is too soft to secure the support of Tory ultra-Brexiteers will also struggle to retain the support of Labour’s Kate Hoey and Kelvin Hopkins, particularly as May’s proposals envisage a continuing attachment to the EU’s rules and regulations, which are the main cause of Labour Euroscepticism.

So May’s hopes rest on Labour’s newest tribe: MPs in constituencies that voted to leave the EU. They fear both the electoral consequences of a soft Brexit and the economic chaos of no deal.

Most high-profile among this group is Caroline Flint in Don Valley, but equally important are the likes of Kevin Barron, the MP for Rother Valley, and Gloria De Piero in Ashfield. Another of their number, Stoke-on-Trent Central’s Gareth Snell, has publicly condemned any Labour MP who would vote against a Brexit deal.

But May’s problem is that this faction is small – probably too small to rescue her. She can pick up the phone in No 10, but there will be no friendly Labour voice at the other end, willing to help her out.