Vitesse Arnhem forward Bony has been linked with a number of Premier League clubs following some superb performances in the Eredivisie, but it is the Swans who are hoping to land his signature.
The Welsh club have identified Bony as one of their top targets as they look to bolster their attacking options ahead of the new season, and Jenkins has previously made it clear that he is prepared to spend significantly to bring in the right players.
It is understood that a deal for Bony could now be imminent, with the Ivory Coast international's agent set for discussions with Jenkins on Monday.
Swansea have already been busy strengthening their squad this summer, bringing in Jordi Amat and Jose Canas and extending Jonathan de Guzman's loan from Villarreal for a second season.
The likes of Newcastle, Sunderland and Arsenal have all previously been credited with having an interest in 24-year-old Bony, who is valued at around £10.2million.