The 29-year-old has been linked with a big money move to the Premiership, with his boyhood club Everton one of many rumoured to be preparing a bid in January's transfer window.
But Jenkins said, "I can assure all our supporters that we will do everything humanly possible to keep Lee scoring goals for our club this season and beyond.
"Lee may love Everton, but Swansea and its supporters love Lee. There lies the difference.
"And who knows? Over the next few seasons we may be playing Everton on a regular basis."