Australian Senator Sam Dastyari, meanwhile, has been evacuated from a restaurant next to where the van ploughed into pedestrians.
An unidentified cab driver speaking to LBC, a local London radio station, described seeing three men jump out of the van.
The agency said a witness had said she saw three people with what appeared to be throats cut.
Horrific scene at London Bridge... bodies strewn over the pavement.
"After that (came) the whole shooting match of ambulances and police". There were also reports of some people being taken into police custody.
Emergency responders on the scene on London Bridge.More news: French Open to start with Petra Kvitova's return to court
"There was reports of gunfire but in fact that could have had more to do with the sheer number of police running into the area", Mr Dastyari said.
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.
I counted five or six people with serious injuries.
"More police came over, I guess they were on radio or something because in around like two or five seconds they shot all the three men down".
Jones added that she witnessed at least four people seriously injured. Police is also responding to an incident in the Vauxhall area in South London, according to the city police.
Police activity spread throughout the Borough Market area.More news: Sonic Mania Will Now Launch On August 15th
A man has been arrested, as Sky News reported live from Borough High Street.
Police later reportedly told media outlets, including the BBC, that there has been more than one fatality. "I froze because I didn't know what to do", Mark Roberts, a 53-year-old management consultant, told Reuters.
"I was told there was something major happening", he said. "I immediately thought this was the Westminster attack all over again".
Transport for London says the busy London Bridge station has been closed at the request of police.
The Associated Press reports that Prime Minister Theresa May called a meeting of the government's emergency response committee.
President Donald Trump tweeted about the incident. A third suspect was believed to still be on the run.More news: Chemists on nationwide strike today over e-pharmacy regulations