The plans come amid increasing concern that the NHS Test and Trace system is failing to reach adequate numbers of people who may be ill in places with the highest infection rates.
The Government has pledged to publish regional data, but has not yet done so.
Thee Scientific Advisory Group for Emergencies (Sage) argues that 80 per cent of close contacts of a positive case should be contacted within 48 hours of a positive result if the system is to work.
However, The Guardian has reported that in Blackburn with Darwen, a unitary authority area in Lancashire where rates of Covid-19 infection are the highest in England, the contacting figure was 54 per cent, and 65 per cent in Leicester, which remains under partial lockdown.
In Luton, which has the sixth highest infection rate in England, just 47 per cent of at-risk people were contacted in time by NHS Test and Trace, according to the reports.
Since the launch of the test and trace system, 169,546 close contacts of people who have tested positive for Covid-19 have been reached and asked to self-isolate.
This is 83.6 per cent out of a total of 202,781 people identified as close contacts.
The weekly figures show 77.9 per cent of close contacts were reached in the week ending July 15, up from 72 per cent in the previous week, but down on the 90.7 per cent reached in the first week of the test and trace system.