The 36-year-old is mulling over his options after ending his 22-year association with Chelsea at the end of last season.
Aston Villa and Birmingham City are two of a handful of clubs interested in the defender, but Jenkins is hopeful that Clement's relationship with the player will help Swansea's case.
Clement is well aware of Terry's qualities having spent numerous years with the centre-back during coaching spells.
"I know Paul has spoken to John and his agent," The Mirror quotes Jenkins as saying. "I don't know what his preferences are because he has options and will go with what suits his family.
"We are well covered in that area and quite strong but I know there is interest with Paul and he spoke with John and his agent himself.
"When Paul knows a players well it is an obvious one for us. Sometimes it works and sometimes it doesn't but we will always try and do that."
Terry opted to leave Chelsea this summer after having limited game time under manager Antonio Conte last season.