PCC David Keane referred the case to the Independent Police Complaints Commission (IPCC) which handed it back.
Mr Byrne, the chief constable since 2014, said he could not comment as he did not know the detail of the claim.
The BBC understands that the complaint originated from an officer with the Cheshire force.
Mr Keane said he referred the allegation to IPCC in the "interest of openness and transparency".
The IPCC returned the investigation for a "local investigation".
Mr Keane said: "This investigation will be conducted independently of Cheshire Constabulary in accordance with the statutory police complaints process."
Father-of-two Mr Byrne, who is a holder of the Queen's Police Medal, previously served as an assistant commissioner for the Metropolitan Police, deputy chief constable of Greater Manchester Police and assistant chief constable with Merseyside Police.
Cheshire's Chief Constable Simon Byrne is being investigated by the county's police and crime commissioner (PCC) over an allegation about his conduct.