The Guardian says Palace made an offer of £22m ($28m) plus £3m in add-ons for the 27-year-old on 29 August, but Liverpool turned it down and asked for £30m.
Sakho, 27, spent the second half of last season on loan at Palace and is reported to be in favour of sealing a permanent transfer to Selhurst Park.
He played eight games for the Eagles after making his debut for the club against Middlesbrough in February and helped them stave off relegation to the Championship.
Palace manager Frank de Boer suggested last week that Liverpool's asking price for Sakho was too high.
"Everybody knows [Sakho] had a major impact last season and, when he's available for us [to buy], I think that is a quality injection for the club," he was quoted as saying by Goal.
"It is not like this now because he is much too expensive for the club."
Eagles chairman Steve Parish said he could not understand the inflated transfer fees paid out by clubs in the current market.
"What's returned to football are the big-money owners, the win‑at‑all‑costs owners, almost bottomless pockets," he said.
"I'd love to see how it all fits with Financial Fair Play, by the way, because I just can't see how half of this fits with that because the revenue of the clubs hasn't grown this year.
"I'm not accusing people of anything. I just don't understand the numbers – for some of the clubs – that they're spending, that are being quoted and how you can make that fit over a period with Financial Fair Play."