Emergency services were called to 10 Mile Lane near Mill Lane at around 4am on Sunday, March 8.
The road was temporarily closed between restaurants including ASK Italian and Las Iguanas while officers dealt with the incident.
A spokesperson for South Wales Police said two men were taken to hospital with non-life-threatening injuries.
Police confirmed that no arrests have been made at this stage, and an investigation is ongoing.
The incident happened during the early hours of Sunday morning when the area around Mill Lane known for its bars, restaurants and nightclubs is often busy with weekend nightlife.
Some people who were in nearby venues at the time said the situation caused concern as emergency services responded in the area.
Officers are continuing enquiries and are urging anyone who witnessed the incident or has information that could help the investigation to contact police.
Anyone with information can contact South Wales Police on 101 or via their website.
