Theresa May is about to make Corbyn an offer on the NHS, apparently as a test of his prime ministerial qualities

'It would be a test of his credentials,' one May ally told me. 'Could he act in the national interest?' Cooperation would be the right thing to do, even if it may seem unpalatable to allow the Tories to neutralise the NHS as an issue

Corbyn's prime ministerial qualities are apparently to be tested, but in some ways he's stuck between a rock and a hard place (PA)

Has Theresa May been watching The Godfather films in her oodles of spare time? Because she may be about to make Jeremy Corbyn an offer Conservatives think he could not refuse – a cross-party approach to the long-term funding of health and social care.

There is a growing consensus among MPs in all parties that the NHS crisis will not be solved by more emergency injections of money and the pressure caused by the rising elderly population can be relieved only by the full integration of health and social care. Many MPs support a one-off tax rise, earmarked for health and care.

May acknowledged the problem by adding social care to the Department of Health’s name in this month’s Cabinet reshuffle. Her aides insist it is more than just a change of nameplate and stationery. Jeremy Hunt, the Health and Social Care Secretary, has said he wants a 10-year funding settlement for the NHS. He is interested in a cross-party approach to take the politics out of health and care, possibly through a royal commission to come up with a long-term plan.

Hunt deserves credit for persuading May to let him stay in his post in the reshuffle. Officials working on care in other Whitehall departments are being switched to health as it draws up a green paper on care, a sign that the Government is serious about integration.

But a royal commission is not the right vehicle. It would take years and be seen as kicking the can down the road. We already know the issues. In 2014, the Barker Commission proposed a single, ring-fenced NHS and care budget, with a single commissioner for local services, and care free at the point of use in the long run. Three years earlier, the Dilnot Commission recommended a cap on someone’s lifetime payments towards their social care.

May’s omission of a cap from the Tories’ election manifesto last June resulted in her laudable attempt to reform social care being dubbed a “dementia tax”, after proposing that people with assets of more than £100,000 and who suffered from dementia would pay for their care, while those with cancer or heart disease would not. Although, May made the fastest U-turn in election history by accepting a cap, the affair contributed to her disastrous result.

Labour suffered a similar fate in 2010, when the Health Secretary Andy Burnham’s sensible plan for a levy on people’s estates after they die to fund social care was ruthlessly branded a “death tax” by the Tories, even though they were supposedly working with Labour and the Liberal Democrats on the issue.

The two episodes highlight the need for a cross-party solution, with the main parties signing up in advance to the findings of a short, sharp inquiry. It could be modelled on the Parliamentary Commission on Banking Standards, an all-party group of MPs and peers which proposed reforms after the 2008 financial crisis.

En route to China, May acknowledged the need to communicate her Government’s policies better, amid growing criticism from Tory MPs over health, housing and education. Although, some ministers doubt her ability to take a decision about anything, there are tentative signs she might indeed make Corbyn an offer on health and care.

“It would be a test of his credentials as a potential prime minister,” one May ally told me. “Could he act in the national interest?”

Corbyn’s desire for a kinder, gentler politics may not extend to hopping into bed with a party he believes is privatising the NHS. And, Corbyn might ask, why should he allow the Tories to neutralise Labour’s most effective weapon? But if Corbyn refused such an offer, to return to The Godfather, might it hurt him in the eyes of voters? Could he really reject the chance to rescue the NHS, in need of some TLC as it marks its 70^th birthday?

Cooperation would be the right thing to do. Corbyn would come under pressure from many Labour MPs to do it; 40 of them were among the 90 MPs to back a cross-party approach last November.

Behind the scenes, Labour figures including Frank Field and Liz Kendall are seeking a consensus with the Tories including Sir Oliver Letwin, Nick Boles and Sarah Wollaston. The go-between is Norman Lamb, the Lib Dem former health minister. He told me: “It is not possible to crack this issue on a single party basis. It involves difficult decisions about how we tax people and what we prioritise in public spending. So any political party steers clear of the difficult judgements and nothing happens. That is why we need to create the space for a solution that sustains our NHS. There is still a chance we can pull this off.”