The Sunday Express says Reds boss Brendan Rodgers is closing in on £20 million Bony as he looks to continue spending the record £75m Luis Suarez transfer cash windfall.
Swansea, who have brought in Lyon striker Bafetimbi Gomis, know they face a fight to hold on to Bony, who they signed for £12m from Vitesse Arnhem last summer.
His contract has a £20m release clause, which Rodgers is prepared to pay his former club.
The Kop boss is also interested in the Swans' £10m-rated left-back Ben Davies.
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