After Mohammed Rehman, 25, discussed possible targets on Twitter under the name Silent Bomber, an investigator contacted him.
Mr Rehman allegedly told him he was planning a martyr operation.
He and his wife Sana Ahmed Khan, 24, both of Reading, deny charges of preparing terrorist acts.
About two weeks before their arrest, Mr Rehman's Twitter account came to the attention of anti-terrorism authorities.
He argued with and taunted other users after asking whether his target should be a "Westfield shopping centre or London Underground".
Mr Rehman allegedly wanted to carry out an attack on the 10th anniversary of the 7 July bombings.
Prosecutor Tony Badenoch QC told an Old Bailey jury an undercover investigator, who used the name Abu Mohammed, then engaged Mr Rehman in a private conversation.
He said: "Silent Bomber revealed his true intentions, asking 'how dumb these Kuffar are lol'."
Mr Rehman also told the officer he was "preparing against them" before his account was suspended, but he reactivated it under slightly different details a week later, the court heard.
"He told the undercover investigator that he was preparing for a Istishaadi (martyr) operation," Mr Badenoch said.
"The level of commitment to an atrocity was perfectly clear, as Mohammed Rehman also asked the undercover investigator if he wanted to join him in a joint operation or whether he preferred to follow the lone-wolf route."
He is also charged with possessing an article for terrorist purposes.
A would-be suicide bomber revealed his plan to target Londoners during an online chat with an undercover investigator, a court has heard.