Last week Badenoch wrote to the prime minister saying he should rule out tax rises or increased borrowing to fund social care reform.
A Conservative spokesman said Andrew would attend in Badenoch’s place and was ready to work with Labour on measures that improved social care – provided they were not reliant on tax rises and borrowing.
“Social care is vital to protecting the most vulnerable in our society,” he said.
“Any changes must be backed by a clear, credible and fully costed plan that delivers better care for the public and value for taxpayers.
“Andy Burnham and Labour cannot rely on vague promises of reform without a clear plan for implementation and detail on what will happen when.”
Sir Ed said on X that he would attend the meeting by Zoom as he was with his family in France and had a disabled son who needed “24/7 care”.
“I’m hopeful this call may prove to be the first step towards fixing our broken care service,” he added.
He has previously said he was ready to do “whatever I can to find a real and lasting consensus on social care” but pointed out there had already been 22 failed attempts at reforming social care since 1997.
“To see the 23rd effort work, we need to do things differently,” he said.
“My party has been pushing for cross-party talks on care for years, because it is the only way to build a solution that actually works.
“For carers and for our country, nothing matters more, so I look forward to meeting the prime minister tomorrow to begin these talks.”
He added his priorities were to put family carers at the heart of any proposals, and ensure no-one lost their home or life savings on catastrophic care costs.
Complaining of the snub, Reform leader Nigel Farage said far from leaving politics aside, Burnham had “chosen to ignore the party that led over 300 national opinion polls and represents thousands of councillors and millions of Britons across the country”.
He accused the prime minister of “conspiring behind closed doors with the failed political establishment”.
Green leader Zack Polanski, asked on Newsnight whether he should have been invited, said: “That’s a case for the prime minister. Actually, I think it’s positive he’s inviting opposition leaders at all.
“I would like to be there, of course I would, I’d encourage the prime minister to extend that invitation… The point here, though, is not politicians’ personal egos, the point is adult social care is a ticking time bomb.”
Adult social care provides state support to help people live more independently, including assistance at home.
Unlike the NHS, social care is not free at the point of use. In England and Northern Ireland, people with savings worth more than £23,250 are not entitled to help with care costs from their council.
In Wales and Scotland, different rules apply in terms of eligibility.