The attack is being treated as a “suspected robbery”, City of London Police said.
Witnesses said two suspects on electric bikes were challenged by brave passersby after they tried to snatch a mobile phone.
The City of London Police said the three victims did not suffer life-changing injuries and were treated at a local hospital.
A statement from Chief Superintendent Steve Heatley said: “Two suspects are being sought as part of the ongoing investigation related to an attempted phone theft and the incident is not being treated as terror-related.
“Stopping phone thefts and bringing perpetrators to justice is a priority for City of London Police.
“The number of phone snatches is down, due to focused effort including working with colleagues in the Metropolitan Police Service last week to make several arrests.”
A nearby shopkeeper said one of the suspects pulled a knife out and “started stabbing” the pedestrians to get away.
He told The Standard: “I was working in the shop and saw a fight between a few people.
“A guy was shouting ‘Oi Oi’ as he passed the shop. He had something stolen from him. A few people tried to help him he looked like an office worker to me.
“I saw two people trying to stop the robbers near the traffic light.
“I couldn’t leave the shop to help. The next thing I heard there was a stabbing. I saw four people lying down on the floor. It was so scary.”
There have been no arrests.
A traffic marshall working nearby heard about the stabbing over his walkie-talkie and rushed to the scene.
“It all happened within a minute and then [the suspect] disappeared”, he said.
“I got there and saw a man on the floor and another person was screaming in agony. The whole public was trying to help with the paramedics and police.”
Police have cordoned off the busy road.
A shopkeeper working nearby told The Standard: “My uncle told me a guy grabbed a phone off a pedestrian and tried to run away.
“Some innocent people tried to stop him after all the shouting. He just took a knife and started stabbing. My uncle saw everything. He is ok.
“The four people were totally innocent just trying to help and then this happened.”
A witness to the aftermath added there was a “fair” amount of blood left on the street as paramedics worked on a victim.
The video below contains scenes which some may find distressing.
Someone thought it was worth stabbing 4 people for a bike on Bishopsgate during this mornings rush hour 😢
But if you listen to @MayorofLondon you’d think climate change is London’s number one issue at the moment 😕#LondonStabbings pic.twitter.com/DvZIBS9255
— #DDD (@DDDaughters) October 6, 2022
She said: “I saw it from a bus. There was a fair amount of blood visible but the casualties were hidden behind a white screen.
“There were lots of paramedics over someone on the floor. At least one person was having CPR being done on them. But I looked away straight away.”
A London Ambulance Spokesman said: “We were called at 9:52am today (6 October) to reports of a stabbing in Bishopsgate, central London.
“We sent a number of resources including three ambulance crews, two advanced paramedics, two incident response officers, a clinical team manager and members of our Tactical Response Unit.
“Our crews treated four patients. We have now taken three of these patients to hospital and one was assessed and discharged at the scene.”