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".

One private hospital told the BBC that 90 people are on a waiting list after it ran out of supplies last week.
Hundreds of thousands of people have signed a national petition for the jab to be offered to all children
It follows a case in England where a mother shared pictures of her daughter dying in hospital from the illness.
Under national guidelines, the vaccine is only available to babies.
All babies in the UK are offered a potentially life-saving vaccine against meningitis B as part of a national immunisation programme.
However, there have been calls for the age limit to be extended to children and teenagers.

The move follows the case of a Thai surrogate mother who said an Australian couple rejected the baby she carried when they found he had Down's syndrome.
The couple, who took home the boy's twin sister, have strongly denied the claim.
Thai police say couples with surrogate children now need a court order before they can leave with the children.
Thailand's military government is rushing through a new law aimed at banning commercial surrogacy.
However, Australian surrogacy agencies believe there are more than 100 couples still awaiting the birth of children to surrogate mothers in Thailand.
A Thai immigration official told the BBC that one same-sex couple with a baby was stopped from leaving Bangkok airport on Thursday because they lacked documentation to prove they were the legal guardians.
Australian broadcaster ABC said three other couples were also believed to have been stopped from leaving Thailand with surrogate babies.
The BBC's Jonathan Head in Bangkok says Thailand has become the favoured destination for many Westerners seeking surrogate mothers.
But while commercial surrogacy is not technically illegal in Thailand, it does violate the code of the Medical Council which regulates doctors and hospitals, he adds.
Earlier this week, Australian couple David and Wendy Farnell denied that they had rejected baby Gammy, now seven months old, when it became clear he had Down's Syndrome.
They said that Thai surrogate mother Pattharamon Chanbua, 21, had refused to hand him over and also threatened to keep his twin sister.
It also emerged that David Farnell had been convicted in the 1990s of multiple sex offences against young girls.
Mr Farnell insisted that Gammy's sister, Pipah, was not at risk of harm from him.
Ms Chanbua, who has two other children, said the couple had asked her to have an abortion when she was told of the baby boy's condition.
She said she refused, as it was against her Buddhist beliefs.
David Farnell denied asking the mother to have an abortion and said that by the time they found out about the baby's condition, it was too late in the pregnancy to abort the foetus.
Moves by a police commissioner to force out her suspended chief constable have been backed by the chief inspector of constabulary, the BBC understands.
[NEXT_CONCEPT]
Private hospitals in Northern Ireland are reporting an increase in the number of inquiries from the public about getting the Meningitis B vaccine.
[NEXT_CONCEPT]
Thailand has imposed new restrictions on taking children born to surrogate parents out of the country.