Wilmington Mayor Dennis Williams said that one of those gunned down was the shooter's wife, the News Journal newspaper reported.
But Delaware State Police Sergeant Paul Shavack would not confirm that relationship with reporters at the scene.
It was unclear whether the assailant took his own life or was shot by police.
"An exchange of gunfire took place, we have these people shot," Shavack said. "I can't give you details of who shot who."
Two police officers were also transported to the hospital with non-life-threatening injuries, he added.
The gunman, said to be in his 20s or 30s, fired shots before he reached the metal detectors in the lobby of the New Castle County Courthouse, Shavack said.
Police did not release the identities of the shooter or victims.
Law enforcement officers at the scene were searching the 12 floors of the courthouse, but said they were confident there had been a lone gunman.
© Thomson Reuters 2013