The 26-year-old Portugal international will undergo a medical in England before finalizing the terms of his transfer.
Luis will join Burnley on a temporary basis with the option to buy the player outright if certain conditions are met.
The details of the transfer include a loan fee of €2.5m with the permanent option set at €22.5m.
Luis's move to Burnley is expected to be the club's final signing ahead of the summer transfer market closing.
Following his departure, Luis bid farewell to Benfica fans after a recent match at FC Alverca.
