The Eagles' latest offer was of £25 million - with £3 million of that fee consisting of add-ons - but the Reds remain intent on pulling in £30 million for the France international.
Liverpool have held out for that price all summer, amidst interest from more Premier League clubs, namely Leicester City and West Bromwich Albion, and remain confident of selling the defender before the transfer window shuts on Thursday night.
It is thought that Liverpool would accept a deal for the ex-Paris Saint-Germain captain that sees structured payments and additional instalments eventually add up to £30 million.
Sakho - who enjoyed a successful loan spell with for the second-half of last season - is keen on a return to Selhurst Park, where he quickly became a fans' favourite for his contributions, helping to Palace to avoid relegation and remain in the top-flight.
Palace remain without a point under new manager Frank de Boer having endured a below-par start - which included a 1-0 defeat to Liverpool at Anfield earlier this month.
De Boer - recently asked about whether he might look to wrap up a deal for Sakho - said: "I don't know. We know what kind of impact he had last season. But, for now, I can't say about it."
Out-of-favour defender has long been likely to leave
The 27-year-old has fully recovered from the knee ligament injury that ended his loan spell with Palace early last term but has not trained with Liverpool's senior squad as a result of his exile.
Sakho has not played for Liverpool's first-team since April 2016 and has no hope of reviving his Anfield career due to his badly-broken relationship with manager Jürgen Klopp.
The Frenchman's Liverpool career will end having made 80 appearances, scoring one goal. His L4 spell ends on a disappointing note, having been an integral member of the squad earlier in Klopp's reign.
But a complex, and wholly unnecessary drug scandal, was the beginning of the end and he has long accepted that his future is away from Liverpool.
Despite his wrongly-imposed drugs ban - UEFA later admitting wrongdoing because the substance that Sakho had used was not on their banned list - the defender earned criticism for not telling Liverpool's medical staff that he was using it.
But Sakho's future was decided when he had to be sent home from Liverpool's pre-season tour of the United States last season for poor conduct.
Voicing his frustrations at Klopp's treatment of him, having returned to full fitness from injury, Sakho was told to look for a new long-term club from early December.
It has taken him until now - despite an impressive spell at Palace - to leave Liverpool, but it makes sense for all parties involved that Sakho complete a permanent move to Palace.