The government announced it had hit its target of 100,000 tests on Friday, but that number has since fallen back.
Speaking at Prime Minister’s Questions, Mr Johnson said “capacity currently exceeds demand” and the government was taking steps to address that.
He said his “ambition” was to hit 200,000 tests “by the end of this month – and then go even higher”.
He also confirmed that he would be setting out plans to begin lifting the coronavirus lockdown on Sunday, adding that he hoped to “get going on some of these measures on Monday”.
The BBC’s health editor Hugh Pym said government sources confirmed that the 200,000 per day target refers to lab capacity rather than individual tests.
Mr Johnson was making his first appearance in the Commons – and his first PMQs clash with new Labour leader Sir Keir Starmer – since the birth of his son and his recovery from coronavirus.
The government must review lockdown measures on Thursday by law – but the PM said he was waiting until Sunday to announce the government’s plans because more data would be available.
And he warned it would be an “economic disaster” to relax the lockdown in a way which triggered a second spike in coronavirus cases.
In March, the PM said the government was aiming for 250,000 coronavirus tests a day but did not put a timescale on that.
Sir Keir Starmer said: “On 30 April, the government claimed success in meeting its 100,000 tests a day target. Since then, as the prime minister knows, the number has fallen back.
“On Monday, there were just 84,000 tests and that meant 24,000 available tests were not used. What does the prime minister think was so special about 30 April that meant that testing that day was so high?”
Mr Johnson replied: “Yes, he’s right that capacity currently exceeds demand, we’re working on that, we’re running at about 100,000 a day, but the ambition clearly is to get up to 200,000 a day by the end of this month and then to go even higher.”
He told MPs that a “fantastic” testing regime will be critical to the UK’s long-term economic recovery.
Labour sources said they planned to hold Mr Johnson to account on his latest testing promise, ahead of a meeting on Thursday between the Labour leader and the PM.
The government believes a track, test and trace programme to quickly identify new cases of coronavirus and prevent the further spread of the infection is the best route out of lockdown.
Sir Keir said the UK now had the highest death rate from the virus in Europe because it had been too slow into lockdown, testing and the provision of Personal Protective Equipment (PPE).
Mr Johnson said it was too early to make international comparisons.