Mr Burnham criticised the policy as "arbitrary" and "the worst of Westminster", arguing it unfairly penalises children.
His intervention is interpreted as a direct challenge to Sir Keir ahead of the Labour conference, amidst suggestions that Mr Burnham is being encouraged by Labour MPs to seek the party leadership.
Senior Labour figures, including Dame Meg Hillier, also support scrapping the cap, highlighting its potential to lift hundreds of thousands of children out of poverty.
Abolishing the cap, which prevents benefits for third or subsequent children, would cost approximately £3bn annually, presenting a significant financial decision for the chancellor.