The 20-year-old has scored 10 goals for the Spanish powerhouses in his first 10 games across all competitions.
Bellingham scored first in the ninth minute and then doubled his team’s advantage in the second half.
Vinicius Junior and Joselu also scored, albeit Joselu missed a penalty in the 84th minute in the 4-0 victory for the hosts.
After excellent work by Dani Carvajal and Luka Modric, his first goal against Osasuna completed a fantastic team goal as he drove the ball into the roof of the net.
After a superb one-two with Federico Valverde, he added his second in the 54th minute with a shot that went between the legs of the advancing goalkeeper Sergio Herrera.
The former midfielder for Birmingham City and Borussia Dortmund received a standing ovation as he departed the field when he was replaced in the 72nd minute.