The Cadogan Square balcony collapsed at about 10am, according to London Ambulance Service.
Tory MP for Chelsea and Fulham, Gregg Hands, tweeted that he spoke to the Cadogan Estate who have said it was not their property involved.
He said on Sky News that the incident was a "tragic accident".
Adding: "They [the Cadogan Estate] tell me the particular property involved had a group of removal workers who were moving a sofa out on the balcony where the combined weight makes it collapse."
It has been reported that both men were workmen.
One man was found dead at the scene and another was taken to hospital.
A London Ambulance Service spokesperson said: “Every effort was made to resuscitate a patient but sadly a man died at the scene.
“Another man was taken to St Mary’s Hospital as a priority.
“Six patients were treated for minor injuries but did not need to go to hospital.”
The man rushed to St Mary's has since died, according to the Metropolitan Police.
Hands suggested that those injured were fellow workmen.
Cadogan Square has been closed to traffic following the incident.
Officials are yet to name the two who died.
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