There are also reports of two armed men entering a bar in the popular tourist city.
Witnesses reported gunshots and seeing people running away and screaming after the crash in the Las Ramblas area.
The El Pais newspaper reported that one person had died and 20 injured and that attackers were holed up in a bar.
The vehicle hit pedestrians in the busy tree-lined promenade in the centre of the city, the Ministry of the Interior said.
The street is one of the most well-known and popular parts of the city, visited by tourists and locals alike.
Images posted on social media appear to show people being tended to as they lay on the street.
The driver of the van is reported to have fled on foot and Spanish authorities have warned people to avoid the area.
Lawyer and University of Glasgow rector Aamer Anwar was walking La Ramblas when he heard screaming.
He said a shopkeeper told him five or six people were badly injured and described the scene as “chaos”.
Mr Anwar said: “I had been to the Cathedral and walking down Las Ramblas for something to eat. Part of it was in the shade so I decided to keep walking down and literally within 10 seconds there was a crashing noise.
“I turned around and people were screaming - I could see a woman screaming with her kids - people started running and jumping into shops. I ran for about 50 or 100 metres and stopped to see what was happening. The police were very quickly on the scene and getting people to move back.
“I could see chaos right at the top area and I spoke to a shopkeeper who had run down and was screaming. He was Bengali so I spoke to him in Urdu and he said a van had driven into a crowd and he thought there were five to six people very seriously injured.”