The audit and inspection was carried out by external regulators in 2008
The Wales Audit Office (WAO) says problems between politicians and management at the top of Anglesey Council are affecting the way the authority operates.
Wales's Auditor General is now being advised to carry out a formal inquiry.
Anglesey council's managing director, Derrick Jones, said he was committed to working to tackle the concerns raised.
The Auditor General is considering the recommendation before deciding whether to carry out an inspection.
The recommendation to the auditor general was made in an annual letter to the council from the WAO.
The letter states that an inspection is needed "due to concerns that difficulties in working relationships between some executive members and some senior officers are having a detrimental impact on the council and its ability to fulfil the general best value duty".
Concerns about member conflict were also raised in last year's annual letter, but at the time the decision was made not to recommend a statutory inspection.
This year's letter goes on to say: "The council does not appear to have unified political and managerial leadership and this is having negative impact on its ability to agree and implement its priorities and to deliver improving services.
"These issues have become increasingly apparent during the preparation of this annual letter."
The council's managing director, Derrick Jones, said there were "a significant number of positive aspects to the report".
"The report highlights the council's good record of achieving its overall revenue budget," he said.
There was also praise for the council's "appropriate arrangement to manage and use financial reserves and satisfactory arrangements".
Mr Jones acknowledged that there are "difficulties" in the working relationships between members of the council executive and directors.
"I acknowledge that this issue needs to be resolved and that a corporate inspection would also allow the authority to move forward," he said.
"I am as you would expect committed to working with the Wales Audit Office to tackle the concerns raised."
Mr Jones added that the issue was "not a reflection on the work of staff", and he thanked them for their continued support in delivering "quality services to the people of Anglesey".
The Wales Audit Office (a merger of the National Audit Office in Wales and the Audit Commission in Wales), has overall responsibility for audit standards across the public sector in Wales.
The Auditor General, Jeremy Colman, is the external auditor of the assembly government and its related public bodies.