A weakened Theresa May has approached the Liberal Democrats to try to secure co-operation in Parliament on a rescue plan for the NHS and social care.

The two parties discussed how to increase desperately-needed funding for the struggling public services, after the Prime Minister lost her Commons majority at the election.

Significantly, the conversation included the Lib Dem plan to increase income tax by 1p for every earner to raise £6bn for the NHS and social care, The Independent has been told.

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“Cross-party working on the NHS, specifically social care, and the Lib Dem plan for a ring-fenced 1p on tax were both mentioned” a source said. “The approach came from the Conservatives, not the Lib Dems.”

Also discussed was Ms May’s once-flagship plan to overhaul mental health laws to end “discrimination and overuse of detention”, which appeared to have been shelved.

Gavin Barwell, the Prime Minister’s new chief of staff, spoke to his opposite number Ben Williams, Tim Farron's chief of staff, late last week.

The move comes despite the Conservatives striking a £1bn deal to be propped up in power by the Democratic Unionist Party (DUP).

That “confidence and supply” agreement only covers key votes on the Queen’s Speech, the Budget, Brexit and security – leaving the Tories without a majority on other legislation.

With 12 MPs, the Lib Dems have a stronger representation at Parliament than the DUP with 10. Ms May fell eight seats short of a Commons majority

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However, a spokesman for the Lib Dems said the party’s co-operation would be restricted to working openly on a cross-party basis, in areas of agreement.

“Tim Farron has been very clear. There will be no coalitions, no pacts, no deals,” the spokesman said.

The Conservatives have strongly denied discussing tax increases, or that they will have to rely on Lib Dem votes for extra funding for the NHS, or social care.

However, since last month’s election shock, the Tories have made clear they are eager to find partners in the Commons to “drive forward the changes our country needs”.

Conservative MPs are desperate to avoid another election, which they fear Jeremy Corbyn would win, while the Lib Dems are eager to show they still have influence.

All parties recognise the need to boost NHS and social care spending quickly, to plug funding gaps running into tens of billions by the end of the decade.

At the election, the Conservatives pledged that NHS spending would be £8bn higher by 2022, but failed to say where that money would come from.

Only yesterday, Health Secretary Jeremy Hunt acknowledged huge problems – admitting there were 2.25m delayed discharges last year, up 24.5 per cent from 1.81m the previous year.

The Independent reported concern after legislation to allow local health chiefs to transform the delivery of care – and find the bulk of £22bn of hoped-for “efficiencies” – was shelved