An injury to a key player means that Real may have to be heavily active in the transfer market in January.
The 25-year-old's current contract expires at the end of the season and he has yet to commit to a new deal.
Given his huge importance to Arne Slot's side, losing Alexander-Arnold midway through the season doesn't seem a feasible option for Liverpool.
Real Madrid have reportedly increased their interest in Trent Alexander-Arnold
Dani Carvajal was taken off on a stretcher during Real's 2-0 win at home to Villarreal on Saturday.
The right-back is set for a lengthy spell on the sidelines after suffering an anterior cruciate ligament injury.
A statement from Real read: 'Following tests carried out on our player Dani Carvajal by the Real Madrid Medical Services, he has been diagnosed with a rupture of the anterior cruciate ligament, a rupture of the external collateral ligament and a rupture of the popliteus tendon in his right leg. He will undergo surgery in the coming days.'
As reported by Sport, Carlo Ancelotti's Real could now look to sign Alexander-Arnold in January.
Alexander-Arnold played the whole game on Saturday as Liverpool won 1-0 away to Crystal Palace courtesy of an early goal from Diogo Jota.
Liverpool remain top of the Premier League and are one point ahead of Manchester City and Arsenal.
Given their push for silverware, it seems unlikely that Liverpool would be open to selling Alexander-Arnold in January.
Real, meanwhile, are in second place in LaLiga behind Barcelona.