Cagliari president Massimo Cellino refuses to rescind Allegri's contract, despite sacking him last season, which means he won't be available for Milan until after June 30.
"The media already knows who will be the next coach," Galliani revealed to Sky Sport 24. "But as he is currently under contract, I cannot confirm anything.
"[Cagliari coach Massimiliano] Allegri is clearly in pole position, though.
"In order to buy a player, we must first get rid of some. In the meantime, we have already substituted [goalkeeper] Dida and [defender Giuseppe] Favalli with [shot-stopper Marco] Storari and [stopper Mario] Yepes.
"In any case, the fans should remain calm as the side can already start to fight for the Scudetto."