But they say they will not be pursuing their interest.
It had been claimed this week the former Leeds player had rejected a move to Turf Moor because he wants to join Newcastle.
A statement read: "Burnley Football can confirm an interest in signing Sheffield United central defender Matthew Kilgallon.
"However, the club is not prepared to meet the agent's excessive wage demands for his player.
"Contrary to a national newspaper report, the club has no intention of breaking its wage structure and disrupting the togetherness of a united squad.
"Burnley Football Club will instead look elsewhere for reinforcements."