Roma president James Pallotta has insisted that he does not want star defender Mehdi Benatia to leave.
Swansea chairman Huw Jenkins has admitted Wilfried Bony could leave the Liberty Stadium if the 'right offer' is made.
The Clarets are the first side to make a move for the 20-year-old former Leicester youngster, while another unnamed npower Championship club have also made an enquiry.
Stags boss Paul Cox told the Blue Square Bet Premier club's official website: "The offer has been turned down as it is well below our valuation of the player. We have also had enquiries from another Championship club.
"We are in the process of putting together a squad which will improve on our position of third place from last season - to do that we will aim to keep our best players.
"As a manager, I would never want to stand in the way of a young player's progression, but I have to do what is in the best interests of this football club.
"I expect more offers to come in for Luke and if the right offer comes along, we will not stand in his way."