Sinclair is poised to seal his move to the Premier League champions in the next 24 hours after weeks of haggling over his transfer fee.
Jenkins is unhappy at how long the deal has dragged on leaving Swansea with little time to bring in a replacement for Sinclair at the Liberty Stadium.
"It has been dragging on a while and I'd have thought if they really wanted to sign Scott they would have dealt with it a bit better than they have, but that's their business," noted Jenkins.
Jenkins is also hopeful of bringing in a couple of new faces before the transfer window slams shut on Friday.
"Hopefully, over the next few days, we'll get a couple more signings in and we can put the transfer window behind us at the end of the month," added Jenkins.
"Our proposed signing of a couple of players is not dependent on Scott's transfer out. I think it's more about numbers and making sure we have the right number in the squad."
The USA will wear their blue away jersey against Jamaica in their World Cup qualifier on Friday night.
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