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.
In other instances, the patient refused to provide details of their contacts.
Some contacts were reached but did not agree to self-isolate, although it is not clear how many refused to comply.
Baroness Harding said the vast majority of people contacted have wanted to comply and provide contacts, but that it is not a "mandatory process".
"These are very good numbers for compliance and we want to encourage people to be part of the system rather than have them be fearful about what might happen to them as they go through it," she said.
She was unable to give a date for when the app will be launched, which will be part of the test and trace programme.
The app has been described as the "cherry on the cake" but "not the cake itself".
A further 151 people have died after testing positive for the coronavirus in the UK, taking the total to 41,279.
The latest figure from the Department of Health is up until 5pm on Wednesday.
It includes those who have died with COVID-19 in hospital, care homes and the community.
The government figures do not include all deaths involving the virus across the UK since the pandemic began, which is thought to have passed 52,000.
The daily figures for each UK nation are:
Scotland - five new deaths - total now 2,439
Northern Ireland - one new death - total now 538
England - 83 new deaths - total hospital deaths 27,790
Wales - six new deaths - total now 1,425
Individual health authorities in the home nations collate their numbers at different times throughout the day and so they may not tally up to the UK government's overall daily total.