From the Super Mario Wiki
Bull's-Eye Banzais, originally referred to as King Bills like the other large versions of the Bullet Bills, such as Banzai Bills and the actual King Bills, are the red heat-seeking variants of Banzai Bills and a larger variety of Bull's-Eye Bills. They attack by homing in on the player.
New Super Mario Bros. Wii
King Bills only appear in World 9-3 of New Super Mario Bros. Wii. Unlike Missile Bills, they only have a limited homing ability where they will track their target's vertical position and then continue forward once they reach that height. Their animation is very limited as well, always maintaining a horizontal position and never turning to point at their target.
The player can defeat it by simply jumping on its head, or by running into it with a Star. Also, a King Bill cannot be frozen by an ice ball; it will break the ice and release itself when the ice ball reaches it. Likewise, POW Blocks are useless against them.
Super Mario 3D World
Super Mario Maker 2
Bull's-Eye Banzais appear under their current name in Super Mario Maker 2. Unlike previously, they can move a full 360 degrees to track players like the Bull's-Eye Bill, and the marks on their eyes are red, like the normal Banzai Bill, instead of black. In a vein similar to Banzai Bills, Bull's-Eye Banzais also receive new designs for all of the present game styles, consisting of a solid red coloration of the original designs. They are replaced by Cat Banzai Bills in the Super Mario 3D World style. Additionally, Bull's-Eye Banzais in the Super Mario Bros. 3 style have extremely subtle differences in their sprites between day and night, with the night ones having one less vertical line of outline pixels behind the domed area.
Mario Kart Tour
Names in other languages