The spokesman said up to 18,000 home testing kits per day will be available by the end of next week.
Another 15,000 tests are anticipated to take place at the drive-through centres on Friday, the PM’s spokesman added.
Number 10 also said the Government is trusting that those applying for tests are key workers, with no eligibility checks in place for online bookings.
The official spokesman said: “As with many other aspects of the coronavirus response, we would expect the public to respond in good faith.
“That is what they have done with other aspects of the scheme, I think we’d expect it to be the same here.”
On testing being rolled out to other groups, the spokesman added: “We want the capacity that we have in the system to be used and you can see this morning that the system is working, people are booking slots and now they are going to be able to undergo tests.”
Asked whether the Government was confident people would be able to test themselves accurately with a kit sent to their homes, the spokesman added: “There are videos available to show people how to do this and people will be given clear instructions.
“We would hope they would be able to do this, yes.”
Under the scheme, test results from the drive-through sites will be sent out by text within 48 hours, and within 72 hours of collection of the home delivery tests.
By mid-morning on Friday, the gov.uk website said it was not accepting any more applications.
It said: “Coronavirus test: applications closed. You can’t currently register for a Covid-19 test. Please check back here later.”
The Department of Health tweeted: “There has been significant demand for booking tests today.
“We apologise for any inconvenience.”