The incident occurred during the second half of Friday's 1-1 draw at Real Betis.
As Alexander-Arnold pressed high up the pitch and failed to win the ball back, Betis broke quickly. The England international jogged back rather than sprinting to recover his position.
Carvajal, watching as an unused substitute, was filmed making a walking motion with his fingers — widely interpreted as a direct dig at Alexander-Arnold's lack of urgency — before leaning over to whisper to a teammate while covering his mouth.
The clip was shared across social media within minutes of the broadcast.
Carvajal has started just three matches since the start of March. His last start came in a 1-1 draw against Girona on April 7. Despite coach Alvaro Arbeloa pledging before the Betis game that Carvajal would receive game time, he did not come on at any point. His contract expires at the end of the season and no extension has been confirmed.
Arbeloa addressed the situation at his press conference ahead of the fixture, pushing back on reports of a rift.
"People are trying to put an unjustified, unfair spin on this. He's more than ready to go to the World Cup. He might be the most important academy product in the history of the club. I've always had a great relationship with him."
On former Liverpool fullback Alexander-Arnold, he said: "Trent Alexander-Arnold is showing a great level. His performance is beyond doubt. Right now he deserves to play. I'm not a coach to gift a minute to anyone."
On Carvajal's future beyond June, Arbeloa added: "All I can say is what I've felt these months, which is that he's helped us. Whatever is good for Carvajal and the club would make me happy."