Stephen Barclay said last night the trial of the app on the Isle of Wight had gone “extremely well”.
Mr Barclay said: “(Matt Hancock) will be making a statement on Monday, (the app) will be rolled out from next week.
“The pilot has gone extremely well on the Isle of Wight.
“One of the questions about the app was whether people would download it – more than half of people with smartphones on the island have now done so.
“One of the other questions was whether privacy was a big constraint but only 2 per cent cited privacy (as a concern) because the data is kept on the phone.”
Mr Barclay said he could not make further comments about when app would be rolled out until the Health Secretary made a statement.
The app could be used to enable stricter lockdown restrictions in some parts of the country while others had more freedom, he said.
“The tracing app, in the future, will enable us to look at where there are localised outbreaks, we do need to look at how we get the most people to download this app as possible, that will enable us to consider what targeted action is taken on a localised basis.”
But Northern Ireland Secretary Brandon Lewis admitted only 1,500 of the 18,000 contact tracers needed as support for the app had been hired.
Mr Hancock has previously promised the contact tracers would be in place by mid-May.
When asked how many had been hired Mr Lewis told Sky News: “I don’t think we’ve got to 18,000 just yet, I think there’s about 15,000 applications, we’re looking to as you say get up to 18,000.”
Pushed again on how many of the 15,000 applicants have been appointed, he added: “As of this morning I’m not sure of exactly how many of the 15,000 have been hired, earlier in the week it was about 1,500, it would have gone up since then.”
A spokesman for No10 said they have now recruited “significantly more” than the 1,500 Mr Lewis mentioned, but would not say how many.
He said challenges with the app, including compatibility problems with some Huawei phones, were still being “ironed out”.
Experts have warned that even if the app does reach enough people, it could fail to pick up half of the cases of coronavirus because it only asks for two of the fourteen possible symptoms.