The Eagles have already brought in Newcastle United winger Andros Townsend for a fee of £13 million, as well as sealing a cut-price deal for Marseille goalkeeper Steve Mandanda.
However, they are in the market for a new quality centre-forward after their goal struggles last campaign, with Benteke having emerged as their new prime target after their failure to land his Belgium teammate Michy Batshuayi.
Palace kicked off negotiations last week with a £25 million bid, but that opening offer was rejected by the Reds as they look to recoup as much as possible of the £32.5 million they paid for him 12 months ago.
According to the Guardian, Alan Pardew's side are now hoping that an improved offer of £27 million upfront which could potentially rise to £31.5 million will be enough to seal the deal, with Benteke thought to be keen to make the move.
Palace have competition for Benteke, with a report last week claiming that West Ham were ready to go toe-to-toe for his signing, but it appears to be the Eagles who are the favourites for his signature.
Benteke endured a difficult season at Anfield last term after his move from Aston Villa, managing just nine Premier League goals in 29 appearances as he failed to establish himself ahead of Daniel Sturridge, Roberto Firmino, and Divock Origi.
However, he could be a superb fit at Selhurst Park, with Palace boasting plenty of talent on the wings to supply the big Belgian with quality service.
Palace will be desperate to get the deal done; they have already lost three strikers this window with the release of Emmanuel Adebayor and Marouane Chamakh and the sale of Dwight Gayle to Newcastle for £10 million.
Their current remaining strikers, Connor Wickham and Fraizer Campbell, scored just five Premier League goals between them last term, with four of those strikes coming from Wickham in the space of four days at the end of February and start of March.