The Sunday Times says the missile veered off course during a test in June last year - weeks before the Commons voted to spend £40bn renewing Trident.
Questioned by Andrew Marr, the PM refused to say four times if she had known about the test ahead of the vote.
The SNP's Nicola Sturgeon called for a "full disclosure" of what happened.
According to the Sunday Times, an unarmed Trident II D5 missile veered off in the wrong direction towards the US - instead of towards Africa - when it was launched from a British submarine off the coast of Florida.
In July - days after Mrs May had become prime minister - MPs voted overwhelmingly in favour of replacing Trident.
During the debate, Mrs May told MPs it would be "an act of gross irresponsibility" for the UK to abandon its nuclear weapons.
MPs backed its renewal by 472 votes to 117. However, all 52 SNP MPs voted against it - as did Labour leader Jeremy Corbyn.
When asked on the BBC's Andrew Marr Show whether she had known then that a misfire had happened, Mrs May said: "I have absolute faith in our Trident missiles.
"When I made that speech in the House of Commons, what we were talking about was whether or not we should renew our Trident."
She was asked a further three times - but did not answer the questions.
The Ministry of Defence did not give details of the test process but said it had been a success.
Scottish First Minister, Mrs Sturgeon - a long-standing opponent of Trident, whose submarines are based at Faslane, on the River Clyde - said the apparent misfire was a "hugely serious issue".
She tweeted: "There should be full disclosure of what happened, who knew what/when, and why the House of Commons wasn't told."
Meanwhile, Mr Corbyn said the reports called for "a serious discussion".
He told Sky News: "It's a pretty catastrophic error when a missile goes in the wrong direction, and while it wasn't armed, goodness knows what the consequences of that could have been."
Nia Griffith, the shadow defence secretary, said it was "completely unacceptable" that Mrs May had "side-stepped" questions.
She called for the prime minister to give "a full explanation" to Parliament on Monday.
Admiral Lord West, the Labour peer and ex-Royal Navy officer, said it was "bizarre and stupid" to not tell anyone about the test.
The Campaign for Nuclear Disarmament (CND) described reports of a misfire as a "very serious failure".
"There's absolutely no doubt that this would have impacted on the debate in Parliament on Trident replacement," its general secretary Kate Hudson said.
A statement issued by both Downing Street and the MoD said the capability and effectiveness of Trident was "unquestionable".
"In June the Royal Navy conducted a routine, unarmed Trident missile test launch from HMS Vengeance, as part of an operation which is designed to certify the submarine and its crew.
"Vengeance and her crew were successfully tested and certified, allowing Vengeance to return into service. We have absolute confidence in our independent nuclear deterrent."
The Sunday Times says the test fire was launched from HMS Vengeance.
It says the Trident II D5 missile was intended to be fired 5,600 miles (9,012 km) to a sea target off the west coast of Africa but veered towards the US.
The cause remains top secret, the paper says, but it quotes a senior naval source as saying the missile suffered an in-flight malfunction after launching out of the water.
HMS Vengeance, one of the UK's four Vanguard-class submarines, returned to sea for trials in December 2015 after a £350m refit, which included the installation of new missile launch equipment and upgraded computer systems.
According to the Sunday Times, it is expected that Defence Secretary Michael Fallon will be called to the Commons to answer questions from MPs.
BBC defence correspondent Jonathan Beale said while the MoD has described the test as a success for the crew and the boat, it has not denied the report that the missile itself might have veered off course.
In the past the MoD has issued a press release and video of successful tests but its silence on this occasion has raised questions as to whether any fault was deliberately kept quiet ahead of the key vote, our correspondent added.
The Trident system was acquired by the Thatcher government in the early 1980s as a replacement for the Polaris missile system, which the UK had possessed since the 1960s.
Trident came into use in the 1990s. There are three parts to it - submarines, missiles and warheads. Although each component has years of use left, they cannot last indefinitely.
The current generation of four submarines would begin to end their working lives some time in the late 2020s.
A guide to the Trident debate
Theresa May is coming under pressure to say whether she knew about a reported misfire of the UK's nuclear weapons system before a crucial Commons vote.