Scotland Yard said its professional standards department are investigating the incident at Chelsea Bridge on Saturday morning.
Officers were called to reports of a male, in his 40s, wielding a screwdriver and shouting at the location just after 9am.
In a video shared on social media, police attended and challenged him on the crossing.
A Taser is discharged but officers were unable to detain the man, who was seen rolling in the middle of the road and screaming.
He gets up, scales the bridge’s edge and hurled himself into river. Scotland Yard said a rescue operation was immediately organised.
He was rescued from the water by the RNLI and taken to hospital, where he unfortunately passed away, the police watchdog said.
The circumstances are now being probed by the force’s Directorate of Professional Standards.
A referral has been made to the Independent Office for Police Conduct.
The IOPC said: “We are investigating contact Metropolitan Police officers had with a man on Chelsea Bridge yesterday morning (Saturday) prior to him entering the River Thames.
“He was subsequently rescued from the water and was taken to hospital in a critical condition. The man, in his early 40s, sadly died in hospital on Saturday evening.
“We were notified by the Metropolitan Police on Saturday and sent investigators to the scene and to the police post incident procedure 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 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. Investigators have overseen forensic examination of the scene.”