The claim came during video testimony from Singapore-born Faiz Abu Bakar Bafana, of Indonesia-based Jemaah Islamiyah.
In testimony recorded in 2002, Bafana said he hosted a guest in Kuala Lumpur in 2000 known only as "John", whom he later identified as Moussaoui during cross-examination.
He said at the time Moussaoui told him of his dream and asked for help in getting flying lessons.
Moussaoui, 37, a Frenchman, was in jail at the time of the September 11 attacks in the US, however prosecutors said his failure to inform authorities of his knowledge of the attack plans contributed to the deaths of 3,000 people.
He pleaded guilty to six charges of conspiracy in April, after being the first person tried in the US in connection with 9/11.
However he has maintained he was involved in a follow-up plot to September 11, not the attacks on New York and Washington.
Testimony
Jurors were played the video showing Bafana's cross-examination in November 2002, about a year after he was arrested in Singapore after suspicions he was plotting to attack a US military base.
In it, Bafana said "John", who he was asked by a fellow JI member to host, told him of his dream.
"He told me he had a dream, flying an airplane into the White House," Bafana said.
He said Moussaoui claimed he told al-Qaeda leader Osama bin Laden of the dream and then asked for help in getting flying lessons, however Moussaoui later deemed the lessons too expensive.
Instead, he busied himself trying to find ammonium nitrate and aluminum powder for explosives.
Bafana said he was later told by "John" that he was leaving Malaysia after a conversation with "brothers" in Europe.
"He said he had managed to get some brothers to assist him for flying training in the US because it would be cheaper in the US," he said.
The US commission which investigated the September 11 attacks found Moussaoui was ordered to undergo flight training in Malaysia in late 2000 by September 11 mastermind Khaled Sheikh Mohammed, but was unable to find a school he liked.
The prosecution later called witnesses from the Oklahoma flight school where Moussaoui arrived in 2001 to learn to fly.
They testified that he was an argumentative student and a poor pilot, who had still not flown solo after 57 hours of instruction.
The prosecution must prove that acts by Moussaoui directly contributed to deaths on September 11 for him to be eligible for the death penalty.
If jurors unanimously agree he is, they will be asked to consider recommending capital punishment.