A sample of households in England, excluding care homes and hospitals, were swabbed to test for current infection.
The ONS says daily cases have risen from an estimated 3,200 to 4,200 since last week.
However there is not enough data to suggest a higher proportion of positive tests in any particular region.
The ONS's estimates of daily cases are higher than those reported by the Department of Health and Social Care because they include people without symptoms who would not otherwise have applied for a test.
Confirmed cases reported by the government for the same period were between 339 and 721 daily over the same period (20- 26 July).
About 350,000 people were newly tested for coronavirus, not including those who were tested as part of the ONS's surveillance study.
These are tests involving a nose and throat swab which can diagnose a current active coronavirus infection, but do not show if someone has had the virus in the past.
Despite the ONS figures suggesting a rise in infections, the official estimate of the virus's reproduction or R number (a measure of whether cases are rising or falling) for England was between 0.8 and 1 as of 31 July.
An R number below one indicates the number of infections is shrinking.
It's calculated using a range of different measures including hospital admissions and deaths.
Because it takes time for an infection to progress to the point of hospitalisation and, in the worst cases, death, there is a time lag involved.
It's possible the latest estimate of R isn't capturing more recent upticks in infection.
The ONS has consistently tested a sample of the population whether or not they have symptoms, so may be better placed to spot a rise in cases in the population at an earlier stage, before they translate to sickness and hospitalisation.
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