Enrique has already confirmed that he will not be at the Camp Nou next season however, after spending three years at the club.
Braida admits that a number of familiar faces have been linked with the job and says Allegri could be in the frame.
“Allegri is a friend and we know each other very well, but it’s really not something we can talk about – it’s all top secret,” he told Radioanch’io Sport.
“There’s a different name coming out every day in the Spanish media, from [Ernesto] Valverde to [Jorge] Sampaoli, from [Ronald] Koeman to Eusebio, who is coach of Real Sociedad and is a part of the Barcelona family.
“Everything is possible. It’s something I would prefer not to talk about.”
Allegri is chasing down a third successive Serie A title with Juventus.