The 26-year-old has been a target for Chelsea over the past couple of months as they seek to add a Premier League proven centre back to their ranks.
According to BBC Sport journalist Sami Mokbel, Xabi Alonso’s side are set to accelerate their interest now that the World Cup is over.
It’s understood that the proposed deal is being handled at ownership level, with latest discussion being described as ‘positive.’
Palace remain keen on keeping Lacroix at the club and would like to extend his current contract, but the player has his heart set on a move away.
The Eagles want a deal worth a minimum £60m for the 26-year-old and have started looking at potential replacements.
Should Chelsea’s move for Lacroix be successful, the future of current defender Trevoh Chalobah, who has interest from Como, will be questioned.