Carvajal was part of the Real Madrid side that recorded a 2-0 win in Saturday's meeting with Villarreal at the Santiago Bernabeu.
However, the Spain international was unable to complete the contest after suffering a serious knee injury in stoppage time.
Real Madrid have since confirmed that Carvajal has "a ruptured anterior cruciate ligament, a ruptured external collateral ligament and a ruptured popliteus tendon in his right leg."
As a result, Carvajal is set to miss the remainder of the 2024-25 season, dealing a significant blow to the European champions.
Real Madrid set to step up Alexander-Arnold pursuit
Real Madrid boss Carlo Ancelotti is now left with the prospect of playing the rest of the campaign with Lucas Vazquez as his only senior right-back option.
While they could turn to the club's Castilla side, Real Madrid may decide to delve into the January transfer market.
According to Marca, the Spanish champions could accelerate their plans to prise Alexander-Arnold away from Liverpool.
Real Madrid were not planning to negotiate with Liverpool because they were willing to wait until next summer when Alexander-Arnold could be available on a free transfer.
However, Carvajal's injury may persuade Real Madrid to test Liverpool's resolve with a January offer for the England international.
Is a January move realistic?
Alexander-Arnold may have entered the final year of his deal, but he remains one of the first names on the Liverpool teamsheet.
The 25-year-old has started nine of Liverpool's 10 competitive matches this season, including all seven of the club's Premier League matches.
Due to Alexander-Arnold's importance, the Reds will be reluctant to part ways with their first-choice right-back in the middle of the season.
However, their stance over whether to entertain a January departure may depend on Alexander-Arnold's contract situation.
If the two parties make progress in contract negotiations then Liverpool would decide against selling Alexander-Arnold in the new year.
A recent report claimed that Alexander-Arnold is to be open to extending his Anfield stay despite the links with Real Madrid.