Police and security services believe the plotters planned to launch an explosive device at Downing Street and then kill May in the ensuing chaos, Sky said.
The director general of MI5, Britain's domestic security agency, briefed cabinet ministers.
Sky said the plot was foiled with the arrest last week of two men by armed police.
London police said that two had been charged with terrorism offenses and would appear in Westminster Magistrates' Court.
It identified them as Naa'imur Zakariyah Rahman, 20, of north London, and Mohammed Aqib Imran, 21, of southeast Birmingham.
Britain faces the most acute threat ever from Islamist militants seeking to inflict mass attacks, often with spontaneous plots that take just days to bring to execution, the head of the MI5 said in October.
After four militant attacks this year that killed 36 people in Britain - the deadliest spate since the London "7/7" bombings of July 2005 - MI5 chief Andrew Parker said the threat was at the highest tempo he had seen in 34 years of espionage.
No 10 Downing Street is the official residence of the prime minister, and heavily guarded. In 1991, Irish Republican Army militants launched a mortar bomb attack on No 10. John Major, the prime minister at the time, was inside but not hurt.
A Downing Street spokesman declined immediate comment on the report.