The arrests in Barking, east London, followed a raid at a flat belonging to one of the three attackers.
A van hit pedestrians on London Bridge at 21:58 BST on Saturday. Three men then got out and stabbed people in nearby Borough Market.
The attackers were shot dead by eight officers who fired 50 bullets. A member of the public was accidentally shot.
The member of the public remains in hospital in a non-critical condition, Assistant Commissioner Mark Rowley said.