The Conservative Party has designated this “family week” in its summer blitz on Gordon Brown. In a speech yesterday, Michael Gove, the Shadow Schools Secretary, said that politics should concentrate on “nurturing relationships” rather than on national spending reviews. He backed support for marriage, an increase in flexible working and more affordable childcare.

Just as the Tories put the concept of general wellbeing above gross domestic product, so they now believe that family partnerships are more important than Public-Private Partnerships. They want to give money to Relate, as well as Network Rail. “Helping families under pressure... to commit and stay committed is one of the most effective anti-poverty, pro-opportunity, pro-equality steps one could take,” Mr Gove explained.

The family is a key part of David Cameron's identity as a political leader. It was no coincidence that he recently invited the television cameras into his home - Brangelina- style - to film him pouring out the children's Cheerios. With journalists, he happily discusses potty-training accidents involving Bob the Builder pants, while veering away from trickier questions about tax cuts.

I remember going to interview him shortly after the birth of his third child - although his aides had insisted that he would not talk about the baby, he rabbited on for hours about nappies but became strangely reticent when asked about Europe. To the irritation of Labour strategists, more voters now know that Mr Cameron is a father than that he is an Old Etonian. Just as they were for Tony Blair, “the kids” are crucial to Mr Cameron's image as Boden Man, the polo-shirted voice of Middle England.

For the Conservatives, the family is a good way of demonstrating a commitment to society, something between the individual and the State - the human face of politics in the “post-bureaucratic age”. It also fits with the “nudge” philosophy that Mr Cameron has espoused - literacy rates will be improved, the Tories believe, if maternity nurses encourage parents to read to their children. The “family values” position of the Tory Right has been rebranded to include creating a better work-life balance and supporting civil partnerships - as well as promoting heterosexual marriage.

The family, in short, is key to the political strategy of the Conservatives between now and the general election. And yet it is the one area in which there is a fundamental disagreement between the two most senior men in the party - Mr Cameron and George Osborne. The Tory leader and the Shadow Chancellor are friends and political soul mates but when it comes to policy on marriage they could be heading for a messy divorce.

From the start of his leadership campaign, Mr Cameron has been committed to reintroducing a recognition of marriage into the tax system. In his first speech as a candidate, the pledge was a strikingly traditional note in an otherwise utterly modernising piece.

Since then, there have been promises to remove the so-called couple penalty in the tax and benefit system too but the Tory leader has always been absolutely clear that if elected to No 10 he will go farther and create a new version of the married couples allowance that Labour scrapped. This is not about supporting parents, or even married parents. Mr Cameron has been categorical that a government run by him would reward people for getting married - those in civil partnerships would benefit as well as heterosexual couples. He cites research showing that almost half of cohabiting couples split up before their child's fifth birthday, compared with one in twelve married people.

Mr Osborne disagrees. For him, it is not the State's job to tell people how to live their lives. He would prefer to use scarce Treasury resources to support parents, whatever family structure they are in, than to reward a childless millionaire hedge fund manager who happens to be married to a lady who likes to lunch. He is concerned that the Tories will alienate voters if they appear to stigmatise single mothers and cohabiting couples. “There is a substantial disagreement,” one insider says. “It's hard to see a way through.”

Although, set against the moral authoritarians in the Tory party, both Mr Cameron and Mr Osborne appear socially liberal - they are completely relaxed about sexuality, for example - the Tory leader is in fact instinctively more traditional than his Shadow Chancellor. When the Commons voted on whether lesbian couples should be able to get IVF, Mr Osborne supported the measure and Mr Cameron did not. The Shadow Chancellor was also uncomfortable with the tone of his friend's recent speech calling for an end to “moral neutrality” - in which the Tory leader said that social problems such as obesity and poverty were the “consequence of choices that people make”. His own discussion of “nudge” politics has concentrated on practical ideas such as telling people how much energy households in their area use.

“George's approach is more pragmatic, less moralising,” says one Shadow Cabinet member. “George is cosmopolitan Notting Hill in outlook, David is more provincial Middle England.” This is partly to do with their backgrounds: Mr Cameron's mother was a magistrate in rural Berkshire and a traditional Tory, Mr Osborne's mother ran a deli in West London and went on protest marches. The Conservative leader is also a regular churchgoer - like Mr Gove he attends St Mary Abbots Church in Notting Hill and not just because it has a good school.

But the reason does not matter. There is now a real tension about a tax policy. Both men hold apparently incompatible positions with passion. And if Labour continues to implode, they may have to decide sooner than they thought how the disagreement can be resolved.

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