Alexander-Arnold was at fault for Lisandro Martinez's opener in Sunday's 2-2 draw against Manchester United at Anfield, falling well below his usual high standards.
The right-back's sloppy performance comes during a long-running debate over his future with the Premier League leaders.
The 26-year-old's contract expires at the end of the season, meaning he can now sign a pre-contract agreement with an overseas club.
Spanish giants Real have reportedly made a bid to sign him but Liverpool are believed to have rejected their approach.
After Sunday's showdown, former United captain Roy Keane took a swipe at Alexander-Arnold's error-strewn display.
"There's talk about him going to Real Madrid -- the way he's defending he's going to Tranmere Rovers after this. He's got to do better," Keane told Sky Sports.
"We talk about how brilliant Trent is going forward, but Trent's defending today. My goodness, it's like schoolboy stuff."
Fourth-tier Tranmere responded to Keane's jibe with their own light-hearted assessment of Alexander-Arnold.
The club posted a picture of their full-back Cameron Norman on X with the message: "Trent to Tranmere Roy? Nah, we're alright thanks".
Former United right-back Gary Neville suggested speculation over Alexander-Arnold's future could have been a distraction.
"The supposed bid from Real Madrid is bad timing for him," he said. "I think that will have unnerved him. Real Madrid are such a powerful club.
"There are three reasons for that performance -- he has played like this before, he just had a bad game, or his mind was affected by all the talk.
"I thought his passing and forward play was the worst I've seen."
Liverpool still hope to persuade Alexander-Arnold to sign a new contract, but he is entitled to leave on a free transfer at the end of the campaign.
Reds defender Van Dijk, who is also out of contract at the end of this season, is confident his teammate can cope with the potentially distracting situation as Liverpool challenge for the quadruple.
Liverpool face Tottenham in London on Wednesday in the first leg of their League Cup semi-final.
"I am fine with it. Trent has his own people around him, his family and we are there as well," Van Dijk said.
"He wants to perform in the best way possible like we all do. That is the main focus.
"In my opinion, maybe ask him, that is the distraction... the game."
