Justine Greening wants to talk about social mobility. If it is not immediately obvious why the Secretary of State for International Development wants to talk about this issue, it becomes clear. Growing up the daughter of a steel worker gave her an insight into what it’s like to struggle, she tells me, when we meet in a conference room overlooking Parliament Square. She says she feels that the Tories are not pushing as hard on social mobility as they ought to be.

Ms Greening thinks the issue needs a champion. She never says so explicitly, but clearly this is her pitch to take on that mantle. As she explains what it was like to be the first person in her family to go to university, how she had to get on in life without well placed connections, and how that helps her understand people’s problems, there is an elephant in the room. The elephant is David Cameron’s privilege. But every time the elephant makes its presence felt, the MP for Putney smiles and politely sidesteps its massive bulk.

She must know that venturing onto the subject of privilege and non–privilege is a risky business. Sir John Major caused a stir last year when he complained that the affluent upper classes still dominated every sphere of British influence, and that hard graft was no longer enough to propel poorer people into positions of power. Major’s comments were seen as a dig at the number of privately educated advisers in Downing Street, the lack of diversity in the coalition cabinet — and at the Eton-educated Mr Cameron himself. After the attack, Cameron promised to redouble his efforts on social mobility.

Nevertheless, Ms Greening feels that the Tories are not pushing hard enough on the issue. More than that, she doesn’t think they will win big until they do. ‘Unless you are pushing it, it will go backwards. Unless we are winning this battle to open up opportunities for young people, the doors have a tendency to gradually close back. This is an agenda that the Conservative party should absolutely own… We should be the people that are pushing forward on it.

‘To my mind the Conservative party has always been most successful when we’ve won the battle for hearts as well as minds and I think that means being a party that can take care of [people’s] dreams as well as their money and help them achieve their goals.’

Ms Greening’s back story is certainly one of the more inspiring ones in the cabinet. She grew up in a working-class family in Rotherham, was educated at a state school and went on to study economics before working as an accountant and financial manager for PriceWaterhouseCoopers, GlaxoSmithKline and Centrica. ‘Both my dad and my granddad worked in the steel industry. The harshest economic lesson I had was the day my dad became unemployed. He eventually found a job filling vending machines. I know what it is like to grow up knowing you are not starting in the best place, or that other people are having a better start than you are. The experience I had growing up, going to my local comprehensive, my family going through difficult times … it’s about understanding what it’s like to start from scratch more.’

There’s the elephant. More than who?

She won’t elaborate. She goes on: ‘My first job was working in Morrisons supermarket in Rotherham.’ I am mindful as she says this that Cameron’s first (and only) job in the private sector was at Carlton television in the 1990s. In their biography of the Prime Minister, Francis Elliot and James Hanning reveal that he was hired after Annabel Astor asked her friend Michael Green, chairman of Carlton, to employ her future son-in-law.

Back to Greening. ‘For me the reason I’m Conservative is because I think that is fundamental to what this party has always been about. Margaret Thatcher’s message to me was, it doesn’t matter where you come from, this is a country where the effort you put in will mean you can get the reward out of it. She was creating a country that was smoothing my path. I could decide how far I got. Even though I didn’t have a whole load of people around me who had already gone to university, it gave me that encouragement to get on.’ So what has happened? ‘Over the years that message has been diluted. My biggest concern is that we are ending up with a country where you have not one ladder to climb up but people are on different ladders. You might start at the bottom of a short ladder that will only get you so high. What we need to recreate is one ladder that everyone can climb up.’

It is a powerful image, all the more poignant, I feel, given Ms Greening’s own progress in government. After luxuriating in the big spending brief of Transport Secretary, with huge infrastructure projects like HS2 to manage, she has been taken out of the limelight somewhat by Cameron, despite being one of the most capable women in the Tory ranks, with extensive business experience. She is gracious when asked about the frustrations of being in government. ‘It is difficult but there are millions of people round Britain doing tough jobs that sometimes they feel are a bit thankless. I’m privileged to be the member of a government. Of course it’s tough but lots of jobs are tough.’

She is careful to say that good progress has been made by the coalition in getting youth unemployment down, in job creation, apprenticeships, welfare reform, and house building. But that is only a start. ‘What is next is really starting to shift the overall culture and mindset. What can we do to identify and bring through Britain’s talent? You see the statistics still there in the proportion of privately educated people in the professions.’

And in Westminster. I ask if she would like to see more people who have worked in Morrisons in the government. ‘Yes, I would. I think it’s really important. One of the reasons it’s important to talk about this is there are actually lots of people in the Conservative party who know what it’s like to start at the bottom… I know how it feels to be slightly locked out of the system.’ It may be the controversy surrounding the reshuffle, in which Greening went unpromoted and other strong women were put in some incongruous positions, but again I feel the poignancy relates to her situation now, as well as then.

But what can ministers like her do to improve the lot of people growing up in less than privileged households, as she did? She suggests tax cuts are part of it: ‘It’s about helping working people get on with their lives and keep more of their money.’

She goes on: ‘I do think as we come out of recession and people look ahead to what kind of Britain is emerging from this, it’s got to be a Britain where our focus is on social mobility. As a party we need to be helping people climb up the ladder. I think the elections we’ve done best on are where people have understood that we know what they are trying to achieve in their lives.’

The inference is that, as yet, the Conservatives are not winning hearts or minds.