Scotland Yard said its professional standards department is investigating the incident at Chelsea Bridge on Saturday morning.
Officers received reports of a 40-year-old man driving a screwdriver and calling the scene shortly after 9 p.m.
In a video shared on social media, police assisted him and challenged him to the crossroads.
A Taser is discharged, but officers were unable to stop the man, who was seen rolling in the middle of the road and shouting.
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He gets up, climbs the edge of the bridge, and jumps into the river. Scotland Yard said a rescue operation had been launched immediately.
He was rescued from the water by the RNLI and taken to hospital, where he unfortunately died, the police watchdog said.
The circumstances are now being investigated by the Professional Force Standards Directorate.
A referral has been made to the Independent Police Conduct Office.
The IOPC said: “We are investigating the contact that Metropolitan Police officers had with a man on the Chelsea Bridge yesterday morning (Saturday) before he entered the River Thames.
“He was later rescued from the water and taken to hospital in critical condition. The 40-year-old man died sadly at the hospital on Saturday evening.
“The Metropolitan Police alerted us on Saturday and sent investigators to the scene and to the police procedure following the incident to begin our investigation.
“Initial information indicates that officers were responding to reports of a disturbance when they encountered the man on the bridge. During the incident, Taser was deployed.
“Investigators have obtained video footage of the police force and the first accounts of the officers involved have been obtained. The used Taser has also been collected and will be analyzed. Investigators have supervised the forensic examination of the scene. “.