Mourinho returned to Real Madrid in June for a second spell after a disappointing campaign last season in which they missed out on the LaLiga title and were eliminated from the Champions League quarter-finals.
"I have always admired the coach (Mourinho). I’ve played against him a couple of times, and it’s a pleasure to work with him and his team," Alexander-Arnold said in a statement.
"It’s intense. The principles and the level of demand are very high, so I’m looking forward to seeing how, the more we get to know each other, the more we learn and the more he can teach us. And we’re all willing and eager to learn and improve.
"I’m sure he’ll teach us many things and help us win trophies this year."
Alexander-Arnold, who joined Real Madrid from Liverpool last year, endured an injury-hit first season in Spain and did not always command a regular place in the starting line-up, rotating in and out of the side.
However, the departure of defender Dani Carvajal in May has opened an opportunity for Alexander-Arnold to establish himself as Madrid's first-choice right-back.
"I’ve been out of action for a long time, so it’s good to finally be back and to lay a solid foundation for a successful season," the 27-year-old Englishman added.
Real Madrid begin their 2026-27 LaLiga campaign against Espanyol on August 22.