The 24-year-old midfielder arrived at Santiago Bernabeu at the start of last season, having impressed for Real Sociedad, and went on to make 29 appearances in La Liga.
Alonso has seen enough to believe Illarramendi can become an integral part of Carlo Ancelotti's squad for many years to come.
"Illarra is the future," Alonso told Cadena Cope radio station. "This is his first year, with big changes.
"It's a giant leap for him. I have no doubts about him.
"Real is like a bull and you have to be ready for it, and Asier will be."
Bizarre Jose Mourinho statement aims to pre-empt punishment after Fenerbahce boss clashes with Galatasaray counterpart
'Andy Robertson has a decision to make as a player, where you either roll your sleeves up and give someone great competition or you accept decisions and say that your time is done.' Alan Shearer on the conundrum facing Arne Slot's Liverpool
