The Reds lost talismanic forward Luis Suarez to Barcelona last week, as the Uruguayan completed a £75m switch to the Camp Nou.
Brendan Rodgers is to be given the funds received to draft in new players, and has targeted Bony as he looks to fill the void in his frontline.
The Ivory Coast international has only been a Swans player for a year, but his 25 goals in all competitions have led to links with both Liverpool and Tottenham.
The front-man himself has revealed that he would consider moving to a club in European competition, which has alerted the Mersesysiders.
It is understood that the Anfield outfit are close to agreeing a deal with the player, with reports having suggested that meetings with his representatives went ahead last week.
Swansea have already signed Bafetimbi Gomis, which could open the door for Bony’s release clause – thought to be around £20m – to be triggered.
Liverpool are also interested in goalkeeper Michel Vorm and toung defender Ben Davies.