The city's metro and train stations have been shut and people have been ordered to stay away from the city centre and stay indoors.
Spanish media cited unnamed police sources who said at least 13 people were killed. Local officials confirmed at least 50 people were hurt, 10 seriously.
No terror group had claimed responsibility for the attack, which was celebrated by ISIS and its supporters on social media.
After the van attack, hostages were taken by at least one armed man at the nearby Luna de Istanbul restaurant, according to local reports.
The El Periodico newspaper had earlier reported gunfire in the La Boqueria food market. Armed police officers have surrounded the restaurant while they negotiate with the attacker.
Witnesses described terrifying scenes as the van went through a red light and mowed down people as they crossed the street and then mounted a pavement to run down more victims, zigzagging at high speed.
The rented van crashed into a kiosk on the boulevard before the attacker fled on foot.
Aamer Anwar was walking down Las Ramblas at the time, which he said was "jam-packed" with tourists. He told Sky News: "All of a sudden, I just sort of heard a crashing noise and the whole street just started to run, screaming. I saw a woman right next to me screaming for her kids. "Police were very, very quickly there, police officers with guns, batons, everywhere.
Then the whole street started getting pushed back." Mr Anwar described the scene as "chaos", with everyone in "panic mode".
He added: "Police officers who got there just started screaming at people to move back, move back.
17/08/17 - 9:15 p.m.
The president of Spain's Catalonia region says police have arrested two people in the van attack in Barcelona's bustling Las Ramblas district.