A management paper on the role of the local government finance director criticises the '10 to 12' councils which have introduced small teams of executive directors without portfolios, with the senior finance officer operating at a lower tier.
Paul Vevers, the commission's director of audit support, said: 'We have grave concerns about this. It is ill-advised not to have anyone with a clear finance brief.'
The commission is happier with other models, providing certain safeguards - such as audit committees - are in place.
Around an eighth of councils have boards which exclude the chief finance officer, but still have a person holding ultimate responsibility for financial affairs, typically a director of resources. About 7% combine the roles of chief executive and finance director.
Mr Vevers said finance directors - qualified accountants or not - needed to embrace a wider role in delivering corporate objectives, rather than just financial administration. He also said s151 officers should be qualified accountants.
Hackney LBC is a leading example of the model causing the commission disquiet. The council's senior team comprises chief executive Tony Elliston and four directors without portfolio.
But Mr Elliston said his management structure had advantages: 'I've worked in traditional authorities as a finance director. The heavy hand of the treasurer acts as a dampener on everything the top team does.'
He said councils with combined chief executives and finance directors were 'in danger of ending up finance-led organisations, rather than outward-looking'.
'Finance directors tend to want to accumulate power, people and resources . . . it's best not to have them at the top table.'
Mr Elliston said when he first took over Hackney, management team meetings were unwieldly affairs attended by 'unreconstructed, unwashed people with vested interests'.
Huntingdonshire DC commerce and technology director Ian Brooke insisted finance directors needed to be at board level, but: '[They] need to bear in mind they should have a wider perspective than book-keeping and number-crunching.'
Surrey CC treasurer Peter Derrick said councils definitely needed a person with direct responsibility for financial matters and that, while it was possible to have a separate s151 officer, it was better to have one person performing both roles.
CIPFA policy and technical director Martin Evans said: 'We would argue that normally the CFO should be a board-level appointment, whilst recognising that . . . they have to earn their status in the corporate management team.'