"The UK is facing an intense threat from terrorism, one which is multi-dimensional, evolving rapidly and operating at a scale and pace we have not seen before", London's Metropolitan Police said on Tuesday.
Rahman, from north London, is also charged with assisting Imran in terror planning, while Imran, from south-east Birmingham, is charged with preparing acts of terrorism.
Prosecutor Mark Carroll told the court Rahman planned to detonate an explosive device at the gates of Downing Street and gain access to May's office in the ensuing chaos and kill her, Reuters reported.
The MI5, Britain's domestic counter-intelligence agency, said nine terrorist plots had been thwarted in the last 12 months, according to two news reports. He said he was of no fixed abode and said his nationality was "Bangladeshi British. Rather it is spreading to new areas, including trying to encourage attacks in the United Kingdom and elsewhere via propaganda on social media", the spokesman also said.
He appeared in court alongside Imran, 21, who is accused of trying to join the Islamic State jihadist terror group by trying to obtain a fake passport with a view to travelling to Libya. Imran said he was "Pakistani British".
Armed police officers stand guard at the gates of Downing Street
A weeks-long investigation into the case culminated last week with the arrests of two men on charges of preparing acts of terrorism, according to the report.
Chief Magistrate Emma Arbuthnot remanded them in custody to appear at the Old Bailey on December 20.
It comes as a new report found that security services could possibly have prevented a suicide attack at the Manchester Arena in May.
Downing Street is protected by armed police officers and separated from the public by fortified gates.
In 1991 the IRA launched a home-made mortar bomb attack on Number 10. The then prime minister, John Major, was inside but was not hurt.
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