Lennon, 30, has not scored a goal in 15 league appearances for the Merseyside club this season and his imminent departure follows the arrival of forward Theo Walcott from Arsenal this week.
The England international, who joined Everton on a permanent deal from Tottenham Hotspur in 2015, missed most of the second half of last season and was detained under the Mental Health Act over concerns for his welfare in May.
The reports said that Lennon has agreed personal terms with Burnley but is yet to undergo a medical.
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