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Naismith has established himself as a key player at Accrington since his arrival from Rangers last summer.
The 22-year-old scored 10 goals last season and his potential has not gone unnoticed among rival clubs.
Bury firmed up their interest with a bid which has been flatly rejected by Accrington.
Accrington boss James Beattie confirmed they had rejected Bury's offer for Naismith and that he hoped to keep the player at the Crown Ground.
"The chairman informed me that Bury did make a bid for Kal, but we rejected it as it was nowhere near our valuation of Kal," Beattie told Sky Sports
"Kal is a valued member of my team and he fits in well with his team-mates.
"He is a very talented player and we don't want to lose him and certainly not at the figures mentioned by Bury."