Figures were up from one in 90 the previous week.
One in 85 is the equivalent of about 658,800 people.
At the peak of the second wave in early January, around one in 50 were estimated to have coronavirus.
Meanwhile one in 20 young people at secondary school in England are estimated to have had coronavirus last week as Covid-19 infection levels remain high in all parts of the UK, the new figures show.
But while the number of people testing positive for the virus is estimated to have increased in England and Wales, it has fallen slightly in Scotland and Northern Ireland.
The percentage of people testing positive for Covid-19 is estimated to have increased in the East Midlands, north-west England, south-west England and Yorkshire and the Humber.
It has fallen in London and north-east England while the trend for all other regions is uncertain.
Yorkshire and the Humber had the highest proportion of people of any region likely to test positive for coronavirus in the week to September 25, at around one in 55.
London had the lowest estimate at around one in 130.
The rate of children testing positive is estimated to have increased among those aged from two to school year 11 and there were also early signs of a possible increase for those aged 70 and over, the ONS said.
The percentage testing positive decreased for those from school year 12 to age 24 and levelled off for people aged 35 to 69 while the trend was uncertain for those aged 25 to 34.
All children aged 12 and over in the UK are now able to have a first dose of Covid-19 vaccine which means those in school years 8 to 11 in England, as well as some in year 7, can get the jab.
The ONS estimates are based on a regular sample of nose and throat swabs taken in the community.
They do not include people in care homes, hospitals or other institutional settings.