The shooter was fatally shot at the scene, the city's police department said, but it was unclear whether from police gunfire or a self-inflicted wound.
The shooter was a current or former employee of the bank, Paul Humphrey, a Louisville Metro Police Department deputy chief, told reporters.
Police said they responded within minutes to reports of an attacker about 8.30am on Monday at an Old National Bank branch near Slugger Field baseball stadium in the city's downtown.
Officers fired at the shooter, police said.
The man was armed with an "AR-15-style" semiautomatic rifle, CNN reported, citing an unnamed federal law enforcement official.
Nine people wounded in the attack were treated at the University of Louisville hospital, a hospital spokesperson said, including two police officers.
One of the police officers was in critical condition, police said.
"We will come together as a community to work to prevent these horrific acts of gun violence from continuing here and around the state," Craig Greenberg, mayor of the city of 625,000, told reporters at a briefing.
Kentucky Governor Andy Beshear, on the verge of tears, said during the briefing that he knew some of the victims.
"I have a very close friend that didn't make it today," Beshear said, "and one who's at the hospital that I hope is going to make it through."
Mass shootings have become recurrent in the United States.
So far this year, the country has experienced 146 mass shootings - using the definition of four or more shot or killed, not including the shooter - according to the Gun Violence Archive, a nonprofit group.
In one of the most recent high-profile incidents, three nine-year-old students and three staff members were killed by a former student at a school in Nashville, Tennessee, on March 27.
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