The Argentine's position at the Blaugrana has come under fire following their Champions League exit in the quarter-finals against Atletico Madrid and their relatively unimpressive Liga campaign.
Barcelona are supposedly ready to part ways with Martino at the end of the season - even though they can still win the league title with a victory over Atletico next week - and Luis Enrique, 44, is one of the leading candidates to take his place.
"Luis Enrique? He is a great coach for Celta," the Barca director was quoted as saying by Sport when questioned about the former Roma coach.
Enrique made over 200 league appearances for Barcelona during his active career and also started his coaching career with the Catalans.
He left Camp Nou for Roma in the summer of 2011 after three years in charge of their B team and has been Celta coach since June 2013.