As to the app, he said the tech is not the only part but "an important part" and the app is working in the Isle of Wight.
But Britain's progress has been criticised: opposition lawmakers said an earlier promise of a nationwide roll-out of a National Health Service (NHS)-developed smartphone app had slipped from the middle of this month.
"We have growing confidence that we will have a test, track and trace operation that will be world-beating and yes, it will be in place by June 1", Jonnson said.
"The track and trace system is going to be ready", Brokenshire told Sky News.
The idea of "immunity certificates" has been shelved for now because of this, although No 10 said it was still exploring it.
When asked directly if the system could work without the app, he said: "Yes".
But Newcastle University's Allyson Pollock, a professor of public health, said the Government's approach had raised numerous questions.
Members of the government's Scientific Advisory Group for Emergencies (Sage) had a special briefing on Thursday on the status of the contact tracing system from Dido Harding, a Conservative peer who is leading the programme.
"They are distinct but complementary, and it is perfectly OK, in fact possibly advantageous, to introduce one before the other", he said. They will be rolled out in a phased way starting next week, he said.
Simon Bryant, director of public health for the Isle of Wight Council, has accepted that Islanders who have had "unclear" Covid-19 test results, or have had to wait up to two weeks for their result, have had their "confidence undermined" by the saga. Relying exclusively on human contact tracing from 1 June was "a big gamble and we may well see an uptick in case numbers".
The app is being developed by a newdivision of the NHS called NHSX with the help of experts from the National Cyber Security Centre (NCSC), who acknowledged that it had been delayed while a number of errors were fixed. These tests indicate if a person now has Covid-19. In the United Kingdom, however, the choice is greyed out, with the text "not available in your region" presented instead. That would mean they could resume social contact with others.