According to the Sunday Times, an unarmed Trident II D5 missile veered off in the wrong direction towards the USA - instead of towards Africa - when it was launched from a British submarine off the coast of Florida.
A senior naval source anonymously told the Times that "there was severe panic that this test launch was not successful", adding that "senior figures in military and government were keen that the information was not made public".
"Ultimately, Downing Street chose to cover up the failed test", the source added. "If the information was made public, they knew how damaging it would be to the credibility of our nuclear deterrent".
When asked about her knowledge of the test on BBC One's The Andrew Marr Show, Mrs May said: "I have absolute faith in our Trident missiles".
The source said an upcoming parliamentary vote on the Trident system on July 18 had made the failure "all the more sensitive".
Prime Minister Theresa May, who succeeded David Cameron on July 13, told Parliament that Britain needed to spend some 40 billion pounds (49 billion dollars) on a new Trident system to deter "serious threats," warning of "a continuing risk of further proliferation of nuclear weapons".
The cause of the failure is top secret but the source suggested the missile may have veered off in the wrong direction towards the US. "We have absolute confidence in our independent nuclear deterrent", he said.
Former Labour defence minister Kevan Jones demanded an inquiry into the claims, telling the Sunday Times: "Ministers should come clean if there are problems".
Given the silence up to now, it is believed that the prime minister and key government officials knew of the test flight failure but kept it from parliament ahead of the debate on Britain's nuclear deterrent.
May was not prime minister during the test, but took office shortly before the vote.
SNP leader Nicola Sturgeon tweeted: "This is a hugely serious issue".
In a joint statement, Mr May's office and Britain's Ministry of Defence said the Royal Navy conducted a routine unarmed Trident missile test in June from HMS Vengeance, as part of an operation created to certify the submarine and its crew. Did you know about it when you were talking in the House of Commons?
MPs are now demanding an inquiry and are calling for a Commons statement from Defence Secretary Michael Fallon on Monday.
Vengeance, the fourth and final Vanguard-class nuclear-capable submarine of the Royal Navy, had undergone a refit in Devonport dockyard before heading out for a firing test to verify if the ship and her crew were ready for active service.
The Sunday Times says the test fire was launched from HMS Vengeance.
The Times said Trident missiles had been test-fired only five times by United Kingdom submarines this century because they each cost £17 million ($29 million).