The France international is a high-priority target as the Blues go in search of a new signing for the centre of their defence ahead of next season.
Crystal Palace are believed to value Lacroix at £60 million and, after early explorations around the potential transfer, Chelsea remain cautious of meeting that figure.
Flashscore sources suggest the ceiling on a deal at this point is closer to £50 million from a Chelsea perspective.
As of yet, no formal talks have taken place, but Lacroix is confirmed by insiders as a player very much in mind at the moment.
It comes as Chelsea are also trying to make advances towards a deal for Granit Xhaka of Sunderland, who have turned down an initial £8 million approach.
Chelsea signed right-back Marco Palestra this week for almost £50 million from Atalanta and are aiming to strengthen the squad further ahead of Xabi Alonso's career as the new boss at Stamford Bridge.
Lacroix is a player they already had on the radar, but Alonso is believed to be on board with the idea of trying to sign him.
Chelsea are trying to show caution in terms of transfer fees at this moment, but Crystal Palace are unlikely to let Lacroix leave for much less than their valuation.
One way around the cost could be to include fringe players in the deal to move in the opposite direction, and that is something that will be mooted.
Meanwhile, Chelsea will also try to move ahead with their deal for Xhaka - but may suffer more frustration.
Sunderland signed the player for £17 million last year, and at this stage, it does not even seem Chelsea would be willing to pay that same amount to take him down to London.
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Dean Jones is a transfer news and football insider for Flashscore, since 2025. He has been a journalist on leading UK newspapers and websites over the past 20 years, establishing himself as an authoritative and trusted voice in the industry.
