Having inherited an "imbalanced" squad when he took over in September, the former England manager wishes to bring the Senegalese to Selhurst Park for a fee believed to be in the region of £15m.
The 24-year-old has allegedly been made the club's priority transfer signing up front, over previous targets Oumar Niasse and Diafra Sakho.
Babacar has only scored four goals in Serie A this season, but all of his 15 appearances have been off the bench as Fiorentina's head coach, Stefano Pioli, has preferred team-mate Giovanni Simeone ahead of the young striker.
He has been with the Italian side since the age of 15 and with hopes to feature more for his national team, a move away from the continent could suit both parties.
The London club are in desperate need of a prolific forward, with winger Wilfried Zaha their current top scorer with four goals, as the team look to avoid being dragged into a relegation battle.
Currently the 'Eagles' sit in twelfth in the Premier League, but are only five points ahead of Stoke City in eighteenth.