The force tweeted a warning telling people in the area to run to safety, hide and then call police if it is safe to do so.
One witness told Reuters that she saw what appeared to be three people with knife wounds and possibly their throats cut. They then responded to reports of people being stabbed in Borough Market.
The Metropolitan Police says armed officers have been sent to the scenes and shots were fired.
Ms Jones said police were on the scene within two minutes, quickly followed by ambulances.
A Downing Street spokesman said: "The Prime Minister is in contact with officials and is being regularly updated on the incident at London Bridge".
"The United States stands ready to provide any assistance authorities in the United Kingdom may request", State Department spokeswoman Heather Nauert said.
London police have confirmed that two incidents at London Bridge and Borough Market were declared terrorist incidents.
"A white van driver came speeding - probably about 50mph - veered of the road into the crowds of people who were walking along the pavement", she told BBC News.
"I could see six people on the floor not moving".
"I heard gunshots, at least two gunshots, and at that point the people who were trying to get out the back entrance came back inside".More news: Doctor in Trump Hotel case released but due back in court
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A large number of police officers were seen gearing up with helmets and riot gear, believed to be on the hunt for suspects. They did not say if officers fired the shots.
The seriousness of the injuries is not yet known, he said. He was at the start of the south side of London Bridge where he told The Guardian he saw a woman being stabbed by three men.
Borough Market, where the stabbing occurred, is a popular market with gourmet food and produce stalls and several restaurants inside.
LANGFITT: Absolutely, and I think that coming - three incidents like this, you know, there was a long stretch where London did not face big sort of terrorist attacks.
Witnesses who were enjoying nights out in pubs and bars in the area were warned to barricade themselves indoors, while people on the streets surrounding London Bridge began to run for their lives.
Medics treated people lying injured on the ground near the market as shocked people cried and shouted around them.
In March, a British convert to Islam ran down people with a vehicle on Westminster Bridge, killing four, then stabbed a policeman to death outside Parliament.
England has been under heightened alert since a bomb was detonated at an Ariana Grande concert in Manchester nearly two weeks ago, killing 22 and wounding more than 100 as they left the venue.
A benefit concert for victims of that attack was scheduled for Sunday in Manchester.