The Ivorian striker will reportedly be sold by the Welsh club if the right fee comes in.
Bony has reportedly been touted by the likes of Liverpool and Tottenham this summer, with his big buy-out clause supposedly putting off both clubs.
But now the Welsh club are thought to have conceded defeat in their attempts to keep Bony, and will no longer hold out for as much as £19 million to sell him.
The report also suggests that Bony has made Swansea aware of his desire to move to a bigger club, leaving Garry Monk with little choice but to offload the unsettled striker.
The Swans paid £12 million for the Ivorian from Vitesse Arnhem last summer, and should still make a profit on the forward despite dropping their valuation.
Liverpool are thought to be in pole position for his signature, with plenty of cash still remaining from the sale of Luis Suarez to Barcelona. Brendan Rodgers also saw a deal for QPR striker Loic Remy fall through last month, and is likely to go in for another attacking option before the season begins.
Bony’s 25-goals in all competitions last season was inevitably going to draw attention to his services, especially since he racked up some European experience in the Europa League as well. A strong World Cup campaign has also seen his stock rise, and with less than two weeks until the season begins, he could leave Swansea looking for a new hit-man.