Dayton won the first-half penalty against Hearts at Rugby Park on Boxing Day to allow Kelly, linked with Celtic, Norwich and West Ham, to slam in the winner with his fifth goal of the season.
The 22-year-old midfielder, whose current deal runs out at the end of the season, recently won his first Scotland cap as a substitute in the friendly win over Luxembourg to further raise his profile, but Dayton wants him to stay in Ayrshire a while yet.
"I would like to see Liam stay. He is a vital player to our team," he said after the 1-0 victory.
"You want your friends to do well in football and go to the next level and I would love to see him go to the next level, but for Kilmarnock football club we want him to be here.
"He deserved international recognition and since then people have spoken more about him, which is good.
"He is a hard-working, strong character with good qualities, he is breaking from midfield and that is what people want, goal-scoring midfielders."