Scotland international Boyd has scored 12 goals for Kilmarnock this season, but is out of contract at the end of the campaign.
The 30-year-old recently stated he is not looking to leave Rugby Park during the January transfer window, but he has been linked with a move to Forest.
Boyd scored six goals in 12 appearances for Forest on loan at the end of the 2010/11 season during Billy Davies' first spell in charge at the City Ground.
Johnston is aware of the speculation, but he has outlined the importance of Boyd staying with Kilmarnock this month.
"Billy Davies had Kris down at Forest but so far we've had no contact," Boyd is quoted in the Sunday Mail.
"We want to keep Boydy because he is a top player. It's not just his goals, he is a big influence to the younger boys. They can see that he has played at the top."