Two men, 20-year-old Naa'imur Zakariyah Rahman and 21- year-old Mohammed Aqib Imran, appeared at Westminster Magistrates' Court in London today charged with terror offences following their arrest by counter-terrorism officers on November 28.
They have been remanded in custody to appear before the Old Bailey court in the city on December 20.
At a brief hearing, the court was told that British Bangladeshi Rahman had "planned to detonate" a bomb at Downing Street gates and in the ensuing chaos try to kill May with a knife.
His terror plot reportedly involved rushing into 10 Downing Street armed with a suicide vest, pepper spray and knife.
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Pakistani-origin Imran is accused of making several attempts to travel overseas to Libya for the preparation of acts of terror, unrelated to the Downing Street plot. Scotland Yard also believe he was trying to join the Islamic State (ISIS) terrorist network.
The latest plot emerges as a new report released this week found that the UKs security services could possibly have prevented a suicide attack at the Manchester Arena in May.
Bomber Salman Abedi killed 22 people when he blew himself up outside an Ariana Grande concert in Manchester.
Barrister David Anderson, who was asked to conduct an independent investigation, said Abedi had been on MI5's radar but that his "true significance was not appreciated at the time".
His report also noted that the Pakistani-origin ringleader, Khuram Butt, of the terrorist attack on London Bridge in June had been on the security services' radar for two years.
The 27-year-old had been watched by MI5.
Butt was involved in "high risk extremist activity", according to intelligence initially received by MI5 but the assessment was downgraded and he was believed to have turned his attention to travelling overseas instead.