Mr Cairns said he had resigned because he had fundamental questions over the leadership of the government. He has been replaced by Ann McKechin.
Ten leading councillors issued a joint statement backing Mr Brown following a meeting with the PM.
They were discussing a range of issues in the run up to next week’s annual conference.
The joint statement said: “There is no doubt that Gordon Brown is the right man to lead Britain.
"With families facing tough times, we need to focus all our energies on helping people manage their finances and keep their homes and jobs secure - that is certainly what Labour councillors are doing, and people expect nothing less from their MPs in Westminster.”
The signatories included Sir Jeremy Beecham, leader of the LGA Labour Group and member of the Labour Party National Executive Committee, and Newham LBC elected mayor Sir Robin Wales.