Sir Thomas Winsor was consulted about Avon and Somerset PCC Sue Mountstevens' plans to "call upon" Nick Gargan to resign.
The PCC says Mr Gargan has lost the confidence of the force at all levels, having been found guilty of misconduct.
But some local Conservative MPs say he has been treated unfairly.
In August, Sue Mountstevens became the first police and crime commissioner to say she would start the process of using new powers to call upon a chief constable to quit, a process which would effectively force him out.
She referred her decision to Sir Thomas - her office has not confirmed his response.
But BBC Points West's home affairs correspondent Steve Brodie said he understood the letter backed Ms Mountstevens' decision. Mr Gargan told the BBC he was unable to comment further.
Now the decision will go to the local police and crime panel for scrutiny, although ultimately, Ms Mountstevens has the final word.
Mr Gargan was suspended in May 2014, amid allegations of inappropriate conduct towards female staff.
An IPCC investigation found those allegations were not proven, but that he had "repeatedly breached data protection principles" and had stored "intimate" material on his police-issue phone.
An independent QC-led panel cleared him of gross misconduct, a sackable offence, but found him guilty of eight misconduct charges and recommended eight final written warnings.
But his three predecessors as chief constable said he should quit for the good of the force.
Announcing her decision in August to initiate the process to call upon him to go, Ms Mountstevens said his position had become "untenable".
But MP for Bridgwater Ian Liddell-Grainger said Mr Gargan was the victim of a "witch hunt", and Bristol North West MP Charlotte Leslie said he had been tried in a "media court".
Moves by a police commissioner to force out her suspended chief constable have been backed by the chief inspector of constabulary, the BBC understands.