He said the government would have recruited 25,000 trackers by the start of next month, capable of tracing 10,000 new cases a day, when the overall number of daily tests would have reached 200,000.
"We have growing confidence that we will have a test, track and test operation that will be world-beating and yes it will be in place by June 1," Johnson told parliament in exchanges with opposition Labour leader Keir Starmer.
He initially told Starmer that he was confident it would be in place by that date if other conditions were satisfied and it was "entirely provisional".