Local MP Nick Raynsford said he had been told the man attacked in the street was a soldier serving at the Royal Artillery Barracks near the attack.
Mr Raynsford said the soldier had been returning to the barracks after a day out when he was attacked.
London Ambulance Service said one man, believed to be the victim of the machete attack, had died in the incident and two others were injured, one seriously.
The BBC reported sources had told them the men were shouting "Allahu Akbar" as they carried out the attack and had filmed carrying it out.
One witness, called James, told LBC radio: "We saw clearly two knives, meat cleavers, they were big kitchen knives like you would use in a butcher's, they were hacking at this poor guy, we thought they were trying to remove organs from him."
"These two guys were crazed, they were not there, they were just animals. They then dragged him from the pavement and dumped his body in the middle of the road.
"They took 20 minutes to arrive, the police – the armed response."
He added: "There was only a few people at first then traffic began to build up because people were getting out of their cars to shout at them they were taking no notice, they were standing there, I think they were proud of what they were doing.
"When they dumped the body in the road, these two black guys had the opportunity to hurt other people if they wanted to because there were brave women with the dead guy on the floor, they were shielding and covering him. The attackers with the knives were standing over these women.
"The guy with the gun, the tall guy with the beanie cap on, even a bus had pulled up – he was going over to the bus and asking people to take his photo."