Chair of the Commons Health committee Jeremy Hunt says local authorities should have the capability to do their own Covid contact tracing.
He made his comments as it emerged several councils in England did not receive some information about how the Indian variant was spreading in their area.
Between 21 April and 11 May, the system only provided details of a limited number of positive cases of coronavirus to the eight local authorities.
On 11 May, they were told by the Department of Health and Social Care that, over that period, 734 positive tests had not been reported.
The number of missing cases was highest in Blackburn with Darwen, Lancashire.
A spokesperson from the Department for Health and Social Care said: “Due to a software issue, there was a delay in tracing contacts of a number of cases. This only affected a handful of local authorities and the issue was resolved as quickly as possible.”