Consultation shows nearly 80% of councils agree the LGMB's training and development should go into an arm's length agency, with policy and governance work taken over by the LGA.
On more detailed proposals, the response is less clear. Nearly 48% say employer functions, including pay negotiations, should stay with a training and development agency while 31% say it should be a separate agency overseen by the LGA.
Quality and management work should go to the LGA, say 54%, while 36% of responses say it should stay with training and development.
The chair of the group leading the review, Greenwich LBC leader Len Duvall, said the board's policy, quality and management work would almost certainly go to the LGA.
Despite responses backing negotiations and training in one agency, Mr Duvall said that was still 'up in the air'. Pay negotiations were very important and 'sometimes the training and development is a bit of a Cinderella and falls off the agenda. It's important to ensure it doesn't'.
Mr Duvall said the review was about better relationships and transparency, and did not show a lack of confidence in the LGMB. Group members were conscious of the effect on staff and wanted to give a clear direction soon.
The review group meets on 15 May to set recommendations for the policy committee.
LGMB chairman Harry Jones also sits on the member group. In its own response, the LGMB argued that only non-employment policy work should go to the LGA, and management, training, negotiations and most of the research should stay together.