The Socceroos legend announced the move on social media, before praising Kyah for "working tirelessly to understand the dedication to get to where he is today."
Cahill will monitor his son's progress with the Under-18s, as the first-team look to return to La Liga after a tough 2019-20 campaign ended in relegation.
"Congratulations my man," Cahill wrote.
"(Kyah) signing for the under 18s academy Leganes. So proud of my son for always doing the small things right and working tirelessly to understand the dedication to get to where he is today."
