Lopetegui replaced Zinedine Zidane as head coach of Los Blancos earlier this summer, with the 51-year-old leaving his role with the Spanish national team to take control of the club.
Carvajal has played under Lopetegui with Spain's Under-19s, Under-21s and the senior team, and the right-back is confident that the pair will enjoy success together in the Spanish capital.
"I have always enjoyed Lopetegui," Carvajal told El Grafico. "I share his ideas of football and he is a coach who gets the most out of each player. He is going to do us well and I am already anxious to put myself at his orders. The bar is now very high but we have to have patience."
Madrid will begin their preparations for the 2018-19 campaign with a clash against Manchester United on Tuesday night.