Atletico were the first to show an interest in the Spain Under-19 and Under-17 forward.
Madrid monitored the 17-year-old's progress last season and have decided to join the race for his signature.
Sources have told ESPN that both clubs see Bravo as a prospect for the future and will not rush him, thus allowing him to go step-by-step and play for the reserves at first before joining up with the first team.
Madrid are confident Bravo will choose them ahead of Atletico, although the player's entourage appreciate the effort made by Atletico after the club increased their offer for the striker.
Whatever the outcome, all parties expect the deal to be finalised in the coming days.
A Barcelona youth product, Bravo joined Leverkusen as a free agent in July 2021.
He became Leverkusen's second-youngest debutant in the Bundesliga at 16 years and 298 days of age in a 1-1 draw with Hertha Berlin last year.