- Video report by ITV News Correspondent Geraint Vincent
The Prime Minister refused four times to deny she knew about a failed Trident test before a Commons vote on the future of the missile system.
An unarmed Trident II D5 missile, fired from the submarine HMS Vengeance, veered off course in June, according to the Sunday Times.
The missile fired from a British submarine off the coast of Florida went towards the US instead of Africa, the paper quoted an unnamed "senior naval source" as saying.
HMS Vengeance, which has just undergone a £350 million refit, is one of the UK's four Vanguard-class submarines.
Mrs May repeatedly failed to answer direct questions on her knowledge of the test, which reportedly took place weeks before MPs approved the £40 billion Trident renewal programme in July.
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.
"When I made that speech in the House of Commons, what we were talking about was whether or not we should renew our Trident, whether or not we should have Trident missiles, an independent nuclear deterrent in the future.
"I think we should defend our country, I think we should play our role in Nato with an independent nuclear deterrent.
The government has publicised past successful tests but in this case remained quiet.
Defence Secretary Michael Fallon may be expected to answer questions about the test.
The paper pointed out the test occurred weeks before MPs approved the £40 billion Trident renewal programme in July.
Kevan Jones, Labour's former defence minister, demanded an inquiry into the claims.
Mr Jones said: "The UK's independent nuclear deterrent is a vital cornerstone for the nation's defence.
"If there are problems, they should not have been covered up in this ham-fisted way. Ministers should come clean if there are problems and there should be an urgent inquiry into what happened."
In a statement, the MoD said: “The capability and effectiveness of the Trident missile, should we ever need to employ it, is 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.
“We do not provide further details on submarine operations for obvious national security reasons.”