Farnsworth received four stitches from the bite after breaking up a fight between the two bulldogs on Tuesday morning.
The wound was in Farnsworth's non-pitching hand.
"He's got stitches in his left hand," Royals manager Trey Hillman said. "He didn't get hurt playing ball. He got hurt at home Monday."
Hillman said Farnsworth should not miss any time with the injury.
"I'm planning on him wrapping it up and having him available," Hillman said.