The veteran striker's 22 goals last season were instrumental in steering the Ayrshire club to Premiership safety, but he was out of contract at the end of the season.
The former Rangers and Middlesbrough striker has been widely linked with a return to Ibrox, but Johnston is not giving up on his most prized asset.
"I spoke with Kris and his agent before he went off on holiday and it was a very positive meeting," he told Sky Sports News.
"He's happy with the on-going 3G pitch project so there is no issue there. He has offers from other clubs and those are better offers financially but that is to be expected.
"He appreciates we can't afford to take financial risks having just resolved a major financial problem.
"He is a hero here, he knows he will be first man on the team sheet so I hope all those factors will lead to him coming to the right conclusion and staying at Rugby Park."
Boyd scored 63 goals over six seasons after beginning his career at Rugby Park in 2000 before enjoying an equally prolific four-year spell at Ibrox. He returned to Kilmarnock in February 2013.