The Welsh club's chairman claimed the club are close to what would be a club-record fee for the Ivory Coast international striker
The 24-year-old finished as Eredivisie's top goalscorer last season with a return of 31 goals in 30 games, collecting the Dutch Footballer of the Year award as the Arnhem-based side finished fourth.
Bony's agent Francis Kacou had expressed frustration that Vitesse’s demands had stalled the deal, while Jenkins hinted slow progress could force the Swans to turn attentions to other targets, however a club-record €16 million bid is now looking likely.
"As long there's not signed, there's no agreement," Huw Jenkins told De Telegraaf. "But it's now up to a few details. We want this finished around [Monday] lunchtime.
"We're talking about the final deals with his manager, but that won't be a problem. He could get his medical in Netherlands and join the team after that."
Michael Laudrup has been busy already this summer, having signed Jordi Amat, Jose Canas, Jonjo Shelvey and Alejandro Pozuelo as well as extending Jonathan de Guzman's loan deal for another season.
However, the Welsh club's pursuit of Bafetimbi Gomis looks set to be over, according to Jenkins, should the Bony transfer be completed due to limited funds being available, despite Olympique Lyonnais having accepted a Swans offer.
"Realistically, it has to be one or the other," he told the South Wales Evening Post. "We had a bid accepted for [Gomis] but, if we sign Bony, it is unlikely we will sign both."