The Independent Office for Police Complaints (IOPC) and the Met’s Directorate of Professional Standards (DPS) are both investigating the incident, which happened at around 9am on Saturday.
A video circulating on social media showed two police officers in a stand off with a man, believed to be in his 40s, on Chelsea Bridge.
Scotland Yard said it had been reported that the man was armed with a screwdriver and “was shouting”.
Footage shows one officer shoot the man with a Taser - who starts screaming and rolling around on the ground.
He then gets up and appears to jump off the bridge while being pursured by one of the officers.
The man was rescued from the River Thames by the RNLI and taken to hosital, however sadly he died on Saturday night.
Scotland Yard said that the man’s family had been informed.
Commander Alexis Boon, from frontline policing, said: “My thoughts are with the family of the man at this very difficult time. I offer my sincere condolences to them for their tragic loss.
“Officers go to work every day to keep the public safe, and so any incident in which a person comes to harm following contact with police is understandably concerning.
“Our officers face some of the most challenging and difficult situations daily, in doing so they are fully aware that their actions should rightly be subject to public scrutiny.
“The Met’s DPS made an immediate referral to the IOPC following this sad incident, and we will co-operate fully with them as they work to understand the full circumstances.”
The IOPC said that investigators have secured police body-worn video footage and initial accounts have been obtained from the officers involved.
The Taser used has also been gathered and will be analysed, while a forensic examination of the scene has been carried out, the watchdog said.
IOPC director Steve Noonan said: “We have spoken to the man’s family to express our sincere condolences and explain our involvement.
“Our sympathies remain with them at this terrible time.
“Our independent investigation is under way into the police actions at the bridge and we have begun gathering and reviewing evidence.”