Officers from the West Yorkshire Police Homicide and Major Enquiry Team (HMET) with a tactical firearms support squad sealed off part of the six-storey Ingledew Court, Moortown, Leeds, shortly before 3pm.
A 20-year-old man was arrested on suspicion of attempted murder and wounding.
The swoop followed a dramatic development yesterday when a 21-year-old was arrested on suspicion of wounding after handing himself in at a Leeds police station. He has been charged and is due to appear in court later today.
Police went to the flats looking for a man in connection with the stabbings in the Call Lane area of the city at about 5am on May 5.
In that incident, four men were knifed shortly after leaving Puro nightclub. Five teenagers arrested initially were later freed on bail. A 20-year-old man has already been charged with wounding and appeared before Leeds Magistrates Court.
In yesterday's raid, some residents were ordered to remain in their flats for their own safety.
Police marksmen with automatic weapons stood in readiness while other police trained binoculars on the flats between Sandringham Way and the adjacent Pepper Hill flats while police dog handlers waited nearby.
Det Supt Bill Shackleton of HMET said: "We mounted an operation to arrest a man wanted in connection with an incident in the city centre on May 5 in which four men received stab wounds"
One woman, who asked not to be named, said: "My son went to go out of his flat and there were officers there and they would not let him out.
"There were also some police stood by the lift so nobody could come out. My son said there were police officers at the rear of the block as well."