A shadow cabinet reshuffle is expected shortly and there are rumours Mr Burnham could be moved on to a different post or demoted.
In the Stafford Hospital scandal this year Tory MPs sought to blame Mr Burnham for not dealing with neglect and poor standards of care that left patients drinking from vases when he was health secretary.
In a question and answer session, Mr Miliband was asked to support his shadow health secretary and his plan to ensure social care, physical treatments and mental health issues were treated in a far more integrated way.
Delegate Margaret Barr, from Eastbourne, asked whether Mr Miliband would endorse a "Burnham plan" for health care.
She said: "We need a new 'B' to Beveridge and Bevan, we need the Burnham plan for health and social care. Will you prioritise this social care initiative? Lots of us want to keep Andy in that post thank you."
But Mr Miliband was noncommittal.