The Northern Irish manager led the Welsh side to the Premier League, and the Swans have been a revelation this season.
The newly-promoted team have played an attacking brand of football, beating the likes of Arsenal and Manchester City along the way.
With speculation that Rodgers could take over at Tottenham should Harry Redknapp take up the England job, Jenkins has confessed a desire to keep his coach at the Liberty Stadium.
“It’s what happens when you do well — players get recognition, managers get recognition,” Jenkins told The Daily Mail.
“Speculation is part of football. I can’t predict the future but we will do all we can to keep our manager and players at Swansea.
“If there is an approach then I will sit down with Brendan and explain as well as I can why he should stay. As a club we can take the dream further. Brendan and the players have had a lot of credit and they deserve it, but I will be doing what I can to ensure no-one leaves.
“The only thing that changes that is not an approach, it is if one of the players or Brendan come to me and say they no longer wish to be here. That’s a different scenario,” he concluded.
By Gareth McKnight