The plot was revealed to the Cabinet yesterday by Andrew Parker, the head of MI5, who also told ministers that security services have foiled nine terrorist attacks on the United Kingdom in the past year.
Sky's Crime Correspondent Martin Brunt said: "It's the latest in a number of terror plots that police and MI5 believe they've foiled this year".
It was not clear on Tuesday night what stage the plot was in or if any suspects have been arrested.
The prime minister's spokesman declined to discuss the details of the alleged plot, but several newspapers have reported that two young men arrested in London and Birmingham last week were making preparations to kill May. One of the men, named as Naa'imur Zakariyah Rahman, 20, is accused of preparing acts of terrorism.
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The revelations come in the wake of a report that showed the Manchester Arena suicide bomb which left 22 people dead in May could have been averted had "the cards fallen differently". The report also showed that MI5 could have stopped the Manchester terror attack with the intelligence they had.
Security measures were introduced in the 1970s but were increased in the 1980s as the threat from Irish republican groups grew. Analysts say the chances of the men gaining entrance to the prime minister's residence would have been remote.
In 1991, Irish Republican Army (IRA) militants launched a mortar bomb attack on Number 10.