Naa'imur Zakariyah Rahman is accused of planning an explosion outside Mrs May's Downing Street office then trying to enter the building with a suicide vest and a knife in a bid to kill her.
State prosecutor Mark Carroll said Rahman planned to detonate an improvised explosive device at the Downing Street gates, then, in the ensuing chaos, try to enter the building "with a view to trying to kill the prime minister".
He planned to use a suicide vest, pepper spray and a knife, Carroll told Westminster Magistrates Court in London.
Rahman had carried out hostile reconnaissance of the area as part of his preparation, it is alleged.
Rahman is charged with preparing terrorist acts and also with assisting another man, Mohammad Aqib Imran, to prepare separate terror acts.
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.
Rahman had two inert improvised explosive devices on his person when he was detained, prosecutors said, while Imran had a copy of a video that had been recorded by Rahman.
The men were arrested separately on November 28.
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Rahman spoke only to confirm his name and date of birth. He said he was of no fixed abode and said his nationality was "Bangladeshi British. I'm Bangladeshi."
Imran said he was "Pakistani British".
They both appeared wearing grey tracksuits.
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Lawyers for the pair gave no indication of their plea on their first court appearance so not guilty pleas were formally registered.
The hearing lasted under 10 minutes.
Chief Magistrate Emma Arbuthnot remanded Rahman and Imran in custody to appear for a short hearing at England's Old Bailey central criminal court on December 20.
Meanwhile in an earlier separate case in the same courtroom, Husnain Rashid, 31, faced charges of helping would-be terrorists prepare attacks, including by sharing a photo of Prince George and his school address.
He allegedly made the social media post about the four-year-old royal along with other messages which allegedly could help others identify UK targets such as stadiums.
Prosecutors also alleged that he planned to join Islamic State in Syria.
Arbuthnot also remanded Rashid in custody to appear at the Old Bailey on December 20.
The court appearances came a day after Home Secretary Amber Rudd told parliament that 22 Islamist terror plots had been thwarted since the killing of a British soldier on a London street by two Islamist extremists in 2013.
