Lewisham asked the Society of Local Authority Chief Executives to propose an independent investigator after the successful candidate, Tony Lear, decided not to take up the post.
Sir Robert will also investigate whether two members of the 24 person appointment panel should have declared an interest as they have business dealings with Mr Lear, who is executive director of environmental health at Brent LBC.
Lewisham's former chief executive Terry Hanafin conducted a preliminary investigation before he left the council into whether members Dave Sullivan and Ian Arnold had a potential conflict of interest.
Mr Sullivan runs a consultancy business which has carried out work for Brent and Mr Arnold is an employee of his company, the Agency for Public Service Development.
Mr Lear told LGC last week he had withdrawn from the Lewisham post because he did not feel he had the confidence of Ms Moran. His decision was unconnected to Mr Hanafin's investigation, he said. Sir Robert is not charging for the inquiry but has invited Lewisham if it wishes to give a donation to a charity of SOLACE's choosing. The council will cover his expenses.
A solicitor by profession, Sir Robert is currently chairman of the Greater Glasgow Health Board. He will report back to Lewisham's policy and resources committee when his investigation is completed.