The disturbance broke out in Walmersley Road, Bury, at 22.10 GMT on Friday.
Five men were taken to hospital, including the three who were stabbed. One was left in a "potentially life-threatening" condition, police said.
Three men, aged 32, 22 and 24, were arrested on suspicion of violent disorder.
Greater Manchester Police said "a number of weapons" including an axe, knife and hammer were found at the scene.
Officers are trying to trace a blue Volkswagen Bora involved in a chase with a silver Seat Leon and a white Hyundai shortly before the fight.
They are also looking for a dark-coloured BMW 5 Series which drove off carrying two white men and two Asian men.
Det Insp Charlotte Cadden said: "We already have a number of witnesses, but we need members of the public to come forward to help us."
Three men were stabbed during a fight involving 10 men and weapons including a knife, hammer and axe in Greater Manchester.