On Monday Justine Greening did something Tory MPs are not supposed to do: she admitted she wanted to be party leader. Somehow she had missed the memo that the more desperately you want the job – and of course you do – the more lavishly you must deny it, offering to sign denials on parchment in your own blood (Michael Gove), or to first be reincarnated as an olive, or locked in a disused fridge, or decapitated by a frisbee (Boris Johnson). In fact this unwritten rule has become so ingrained in party lore that when Ruth Davidson categorically ruled out ever becoming prime minster in September this year, speculation immediately began over when her leadership bid might start in earnest.

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Yet here was Greening on the Good Morning Britain sofa, telling Susanna Reid she “might be prepared” to stand for the leadership. As an astonished Reid exploded with gratitude, there was a second or two where Greening herself seemed a little stunned at her own candour. But then Greening – once described by this newspaper as “the Tory party’s other blonde” – has been privately letting her ambitions be known for quite some time. This time last year, for example, a quote from an “ally” ran in a Sunday paper saying she would “run if there was a vacancy”. Calculated or not, she meant what she said.

This kind of talk from an outspoken people’s vote supporter, on the morning of Philip Hammond’s budget, could have been custom designed to get Brexiteer backs up. “There is more chance of Mr Bean becoming King of England,” one indignant right-leaning Tory party member told me. And it is true that Greening has next to no hope of becoming leader should May’s premiership end before Brexit is completed. The membership is allergic to the idea of a people’s vote, and moreover Greening represents the left of the party: while Brexit is under way – and unless a second referendum is announced – a replacement for May will be drawn from the centre or the right.

But beyond that date, Greening’s credibility rises. The hope among many Tory MPs is that they can distance their party from Brexit once it is over, and at the centre of this idea is the coronation of some fresh candidate, untainted by the whole business. Greening fits this portrait: she resigned as education secretary earlier this year, she hasn’t faced the sticky choices that are part of being a cabinet member during the Brexit years. She supports a second referendum, and she has cabinet experience, unlike some of the other names that have been floated (Tom Tugendhat, Johnny Mercer).

She ticks other boxes too. She is from Yorkshire – fairly unusual for a Tory. She’s been banging the drum for social mobility for ages (even while at the Department of International Development, where it was hardly her brief) – a cause other members of her party are now starting to take on. And as a former accountant she has a good brain for detail, and numbers. “It’s a real pain in the neck in meetings: she’ll be able to quote lines from obscure research documents,” says one party supporter.

True, she is pretty dull: just about the only memorable thing she has done is announce she was in a same-sex relationship two days after the 2016 referendum. But if she’s Maybot 2.0, this version seems to have at least got the major bugs worked out. Here, for example, is a typical interview answer about her hobbies – to compare and contrast with Theresa “fields of wheat” May. “I don’t have any strange hobbies, or a weird background. I am certainly not from a privileged family. Maybe I need to invent something to make myself more interesting – Chinese knife-throwing or something.” Boring, yes, but not totally embarrassing.

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If Greening does eventually stand, her biggest rival will be Amber Rudd, who hails from the same left-leaning, remainer section of the party. Both have experience in big jobs, both are clinging on to their seats by wafer-thin majorities (Greening’s is 1,554), both have a reputation for seriousness. But Rudd has recently started to soften her image – opening up, charmingly, about her personal life in interviews, talking about her sacking, and showing a little personality. That should be Greening’s next move.

• Martha Gill is a freelance political journalist and former lobby correspondent