According to Sky Sports, the Premier League new boys have agreed a deal in principle to sign the highly rated 26-year-old.
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It’s understood the deal will be a season long loan with an obligation to buy Luis for a fee believed to be around £21.6 million.
There is yet to be an agreement between Burnley and the player, however, and it remains to be seen whether he would be open to the move.
Luis played a key role for Benfica last season, scoring two goals and popping up with one assist in his 43 games across all competitions.