The 6-foot-3 Griffey had 23 receptions for 382 yards and two touchdowns last season with Arizona.
His best season was in 2014, when he had 31 catches for 405 yards. The Colts, who already feature T.Y. Hilton, Donte Moncrief and Phillip Dorsett at receiver, didn't select any players at that position during the draft.
Ken Griffey Jr., who was inducted into the Hall of Fame in 2016, played 22 seasons in the majors with the Seattle Mariners, Cincinnati Reds and Chicago White Sox. He finished his career with 630 home runs, 13 All-Star appearances and 10 Gold Glove Awards.
Trey Griffey was selected by the Mariners in the 2016 amateur draft in the 24th round -- fittingly, his father's old team and jersey number. The pick on the draft's final day appeared to be the Mariners paying homage to their former star, as Trey Griffey hadn't played baseball competitively since before high school.
"I told them from day one that I was pretty much done with baseball,'' Griffey told FanRagSports earlier this year. "I told them, 'My love is football. I'm not focused on baseball. I'm focused on football.'''
The Associated Press contributed to this report.