Sir Keir Starmer has two weeks to come up with a plan for social care, a former Labour health secretary has warned.

Andy Burnham, who published his own plan to overhaul the system over a decade ago, said the Labour leader must outline his idea by the party’s conference at the end of September.

It follows days of criticism of Sir Keir from within his own ranks, amid frustration that Labour has not offered an alternative to the government’s social care proposal.

On Wednesday evening the party voted against plans to increase National Insurance contributions to raise extra funding for the NHS and, eventually, social care.

Speaking at Prime Minister’s Questions Sir Keir said his party would ensure those with the “broadest shoulders” would bear the brunt of the cost of the new system and criticised the government for increasing taxes for working people.

But when asked for his full plan the Labour leader could not give more detail, prompting frustration from some of his MPs who feel the failure is a missed opportunity.

Mr Burnham, who is now the mayor of Greater Manchester, told Sky News: “It frustrates me that Labour didn’t back my plan years ago, I’ll be honest about that, that did frustrate me at the time.

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“But with regard to now I don’t think the Labour party can come up instantly with its alternative to what the government announced yesterday, it needs to digest what the government has said, but I would like to see them do it soon – perhaps at conference.

“We have a government plan here, I think they’ve gone down the wrong path because they’ve loaded the whole cost of social care on the shoulders of younger people, lower paid people, people who have student debt, people struggling to get on the housing ladder, I don’t think that’s fair.

“There’s an opportunity here for Labour to set out a much better alternative and I would say to them I think they should do that sooner rather than later, but the conference would be a good time to do that.”

A number of Conservative MPs rebelled against the plans and more abstained, citing concerns similar to those raised by Labour about how the plan would work and how much it will cost working people.

The social care proposals will become policy however, as enough MPs backed Boris Johnson, some despite their concerns.

Sir Keir is preparing for what could be a turbulent party conference amid disagreements over policy and a row with Labour’s youth wing.

The annual event, which was cancelled last year, will take place in Brighton at the end of September.

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