Two men were taken to hospital after the altercations in the Frodingham Road area at around 1pm yesterday (Tuesday).
A 20-year-old man was taken to Scunthorpe General Hospital and a 29-year-old was taken to Hull Royal Infirmary - both having sustained serious non-life threatening injuries.
A third man was seen running from the area and officers are appealing for help in tracing him.
He is described as white, aged between 18 to 24, around 6ft tall, of slim build, with short, brown hair in a crew cut style, wearing a blue hooded top, blue jogging bottoms and black trainers.
The incident took place in the area between Porter street and Percival street.
Detective Chief Inspector Bob Clark said: “We are treating this matter extremely seriously and have a large number of resources in the area carrying out enquiries to locate and arrest the third man involved in the altercation.
“This was a contained incident involving three people who were clearly known to one another. As a consequence the outstanding man is not believed to pose any danger to the wider public.
“We appreciate the community will be alarmed by the incident and we will be providing reassurance and support with extra high visibility patrols in the area.”
Anyone with information is asked to contact Humberside Police on 101 quoting log 185 of 22/09/15.