The former Liverpool star lasted just four minutes of Tuesday night's 2-1 victory over the French side as the record 15-time champions opened their continental campaign with a hard-fought victory.
That was despite being reduced to 10 men when Dani Carvajal, who had replaced Alexander-Arnold, was dismissed with 18 minutes left.
"After the tests conducted today on our player Trent Alexander-Arnold by Real Madrid's medical service, he has been diagnosed with a hamstring injury in his left leg," the club said in a statement.
Local press reported that Alexander-Arnold would be out for about three weeks.
He was signed in the close season to provide competition for Carvajal, who suffered a long-term injury last season.
Both right-backs will now miss Real's second Champions League encounter away to Kazakhs Kairat Almaty on September 30th.
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