The head of the test and trace programme says the vast majority of people have wanted to comply and provide contacts.
Samples are taken at a coronavirus testing facility in Temple Green Park and Ride, Leeds
One third of people who tested positive for coronavirus could not be reached by the government's contact tracing system, or refused to hand over their contacts.
Between 28 May and 3 June, 8,117 people who tested positive for COVID-19 were transferred to the NHS Test and Trace system.
Around two-thirds (5,407) provided details of people they had come into close contact with to the government scheme, the Department of Health has said.
But 2,710 did not provide information about their contacts or could not be reached.
Some 31,794 contacts were identified by the system and 26,985 (85%) were able to be contacted and told to stay indoors.
Contacts include people whose cases are currently being managed as part of active outbreak investigations.
The test and trace system is believed to be key to helping ease lockdown measures.
Baroness Harding, the head of the test and trace programme, admitted it was not yet at the "gold standard" but said it was now a "functioning service" and would improve over the summer.
According to the data, 79% of people who tested positive for the virus were reached within 24 hours, and a further 14% were reached within 48 hours.
A number of contacts were not able to be identified as the contact details for some of those who tested positive for coronavirus were unavailable or incorrect, or they did not respond.