In the week that Alexander-Arnold announced he would be leaving Liverpool this summer, the fullback was a second-half substitute in Sunday's 2-2 draw with Arsenal.
And when he entered the pitch, Alexander-Arnold was greeted by a chorus of boos.
Afterwards, Carragher said on Sky Sports: "I think the manager will be shocked by how vociferous the reaction wa.
"That might be his last game for Liverpool but, if frustrated with Trent, sing Bradley's name, sing Gerrard's name. Don't boo your players because it opens you up to criticism.
"Liverpool supporters think playing for Liverpool is the utmost and when you're a local player, like Trent is, like I was, you can't see yourself as bigger than the club.
"Supporters of any club wouldn't want to hear this, but most players in that dressing room probably want to play for Real Madrid.
"I've been in that dressing room, with players who have left us for Real Madrid or Barcelona.
"It's sometimes really hard for supporters. You feel like they fall for it every time, that a player loves them and then moves on."