Some of the unreported cases were then added to Saturday's figure of 12,872 new cases and Sunday's 22,961 figure.
It said all of the cases "received their Covid-19 test result as normal".
BBC Health Editor Hugh Pym said it meant the contacts of those who tested positive had not been approached.
Public Health England said the cases were missed off daily reports between 25 September and 2 October.
It added the IT problem had been resolved and all cases had now been handed over to the test and trace system.
Labour's shadow health secretary Jonathan Ashworth said: "This is shambolic and people across the country will be understandably alarmed.
"(Health Secretary) Matt Hancock should come to the House of Commons on Monday to explain what on earth has happened, what impact it has had on our ability to contain this virus and what he plans to do to fix test and trace."
Earlier, Prime Minister Boris Johnson warned it may be "bumpy through to Christmas" and beyond as the UK deals with coronavirus.
Speaking to the BBC's Andrew Marr, the PM said there was "hope" in beating Covid, and called on the public to "act fearlessly but with common sense".
At a time when the testing system has come under intense scrutiny after reports of delays and a system struggling to keep up with demand, the latest revelation could not have come at a more awkward moment for the government at Westminster.
Because the nearly 16,000 extra positive test results had been not entered into the test and trace system, their recent contacts were not immediately followed up.
Experts advise that ideally contacts should be tracked down within 48 hours.
Officials say the technical problem, thought to be IT related, has been resolved, with all the new cases added into totals reported over the weekend.
But all this will hardly improve public confidence in the testing system in England.
And it muddies the waters for policy makers and officials trying to track the spread of the virus at what the prime minister has called a "critical moment".
On Sunday, a note on the government's coronavirus dashboard said: "The cases by publish date for 3 and 4 October include 15,841 additional cases with specimen dates between 25 September and 2 October - they are therefore artificially high for England and the UK."
Public Health England's interim chief executive Michael Brodie said a "technical issue" was identified overnight on Friday, 2 October in the data load process that transfers Covid-19 positive lab results into reporting dashboards.
"After rapid investigation, we have identified that 15,841 cases between 25 September and 2 October were not included in the reported daily Covid-19 cases. The majority of these cases occurred in most recent days," he said.
"Every one of these cases received their Covid-19 test result as normal and all those who tested positive were advised to self-isolate."
He said they worked with NHS Test and Trace to "quickly resolve the issue and transferred all outstanding cases immediately into the NHS Test and Trace contact tracing system".
"We fully understand the concern this may cause and further robust measures have been put in place as a result," he said.
Test and Trace and Public Health England joint medical adviser, Susan Hopkins, added that "a thorough public health risk assessment was undertaken to ensure outstanding cases were prioritised for contact tracing effectively".
A Department of Health spokesman earlier said the issue had not affected people receiving test results, and all those who tested positive had been informed in the normal way.
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