The Met has been contacted for information about the third patient.
Following a pursuit a 17-year-old boy was arrested by officers on suspicion of attempted murder.
He remains in police custody.
Old Kent Road eastbound remains closed and officers have urged people to avoid the area where possible.
A spokesperson for Metropolitan Police said: “If you have any information that could help police, please call 101 quoting CAD 8387/27AUG.
“If you wish to remain anonymous, please speak with Crimestoppers on 0800 555 111.
“Old Kent Road eastbound remains closed at this stage. Please avoid the area where possible.
“We understand the concern this incident will cause within the local community.
“Residents can expect to see an increased police presence in the area over the coming days.”
A spokesperson for London Ambulance Service said: “We were called at 10.44pm on Wednesday 27 August to reports of a stabbing on Old Kent Road, SE1.
“We sent resources to the scene, including ambulance crews, an advanced paramedic, a paramedic from our tactical response unit and an incident response officer.
“We also dispatched a trauma team from London’s Air Ambulance, which consists of a doctor and a paramedic in a car.
“We treated three patients for stab wounds and took them to major trauma centres.”
