The 20-year-old made six first-team appearances for Los Blancos during the 2015-16 campaign, before spending last term on loan at Bundesliga outfit Wolfsburg.
Mayoral only managed two goals in 21 appearances for Wolfsburg last season, but the Spain Under-21 international is desperate to prove his worth at the Bernabeu.
"Of course I would like to stay here as I'm at the best club in the world," Mayoral told Marca. "I must speak with [Zinedine] Zidane to see what he believes, but I am happy to be here.
"There is only one centre-forward here (Karim Benzema), so I believe there is a chance. If I have to leave I want to stay in Spain, I am thankful that the club have shown confidence in me after a year abroad which was not very good."
Mayoral joined Real Madrid from local club Parla in 2007.