McKenna asked to leave at the end of last season and attracted interest from Aston Villa, Stoke City, Nottingham Forest and QPR in the summer but Aberdeen were reluctant to let him leave and wanted around £6million for the 23-year-old Scotland international.
Burnley have long monitored his progress and he also retains admirers in the Championship.
At Pittodrie, McKenna is under contract until 2023.
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