The Toffees have been tipped to sign another right-sided defender as back-up to Seamus Coleman which would push Kenny further down the pecking order.
As a result, Clarets boss Sean Dyche is keen to bring the young defender to Turf Moor on a season-long loan, according to Football Insider.
The 22-year-old made just eight Premier League starts last season as he struggled to dislodge Coleman from the starting line-up, despite the Irishman’s indifferent form during the autumn.
Marco Silva revealed last week that he was planning to hold talks with Kenny over his future, saying that he may seek to send him out on loan.
His decision to play Kurt Zouma at right-back ahead of Kenny on the final day of the season against Tottenham was particularly telling and suggests his future lies away from Goodison Park - for now.
Should Everton recruit a new right-back then a move to Turf Moor and the chance to play regular first-team football would make perfect sense for everyone involved.