The incident happened at around 4.10pm on Monday, November 24 off Oakhurst Way, in Redhouse, near a park.
The man has been taken to hospital with suspected life-changing injuries.
The suspects were also riding electric bikes.
Wiltshire Police have since authorised a Section 60 order in Swindon giving officers temporary additional powers to stop and search people who are suspected of involvement in criminality and disorder in the Redhouse area.
They were authorised at 7.25pm on Monday and will be in place for up to 24 hours, and will be kept under constant review.
A Section 60 order comes under the 1994 Criminal Justice and Public Order Act and allows police officers to stop and search anyone within the designated area without needing reasonable grounds.
The Section 60AA order further extends these powers, enabling officers to require individuals to remove face coverings that may obscure their identity.
Superintendent James Brain said: "We understand how concerning incidents like this are for the local community and we want to reassure residents that we are fully committed to tackling violent crime and keeping our streets safe.
"These orders were authorised at 7.25pm following a serious stabbing in Swindon this afternoon, granting officers temporary additional powers to stop and search individuals in the area, helping us to prevent further violence and identify those responsible.
"This does not mean people will be stopped indiscriminately and officers will still follow usual stop and search processes.
"Residents in the local area will likely see a heightened police presence in the area while officers from our CID conduct an investigation into the stabbing.
"We are keen to hear from anyone who witnessed the disorder or has any dash cam or doorbell footage which may have captured the offenders."
Anyone who can help should contact 101 quoting log 241 and the incident date (24/11).
Information can be left anonymously with Crimestoppers on 0800 555.