The Iceland international has been a revelation since joining the club on loan in January and the Liberty Stadium chief is ready to fight to extend his stay in south Wales
The 22-year-old has taken the Premier League by storm since arriving from Hoffenheim in January, notching seven goals in 15 appearances for the Welsh outfit.
Although the player himself recently admitted that he would like to remain with the Swans beyond this season, the chairman is predicting a tough battle to keep the Icelander at the Liberty Stadium, with the likes of Arsenal having been linked with his signature.
Jenkins told BBC Radio Wales: "He's done a great job for us and is still only young.
“Our aim is to keep our best players. We've always tried to do that over the years and it will be no different with Gylfi.
"The fact is when you take a player on loan, control is out of your hands. They [Hoffenheim] wouldn't commit to do a deal before we took him, which we would have loved to have done.
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"No doubt there's going to be a bit of a battle on our hands to secure his services for next season. The other thing is they [Hoffenheim] have had a change of manager as well and he's still contracted there for a couple of years."
Swans manager Rodgers recently admitted that he would love to keep Sigurdsson at the club, having worked with the player previously when he was in charge of Reading.
Meanwhile, Jenkins is also hoping that the club can also recapture defender Stephen Caulker from Tottenham.
The youngster has impressed alongside Ashley Williams at the heart of the Swans' defence during his own loan spell this season.