Madrid directors will meet 24 hours after watching the side capitulate at the Nou Camp, losing 5-1 and sliding to ninth in the table.
Conte is the overwhelming choice to replace the current Real Madrid coach, who was left without the support of club president Florentino Perez on Sunday.
Perez is understood to have left the Camp Nou without speaking to his coach or going down to the dressing room to face the players.
Real Madrid understand Conte is due back in Europe on Monday after family holidays in Africa.
They could opt to send a caretaker coach to the cup game with a view to having a new man in charge for the visit of Valladolid next weekend.
Current B-team coach Santiago Solari could step in for the game against Melilla although he does not have the backing of the first team squad to take the job long term.
Real Madrid have picked up one point from their last five league games and Spanish paper El Mundo reported on Sunday night that such is the disarray at the club that Lopetegui’s chief analyst Antolín Gonzalo, seated close to the press box at the Camp Nou, could be seen banging his head against his laptop in the first few minutes of the game.