The 29-year-old is set to return to his hometown club in a move worth in the region of £10million.
It is understood the deal is fully agreed with the player having not travelled on West Brom's pre-season trip to Spain to complete the move.
Rodriguez began his career at Turf Moor back in 2007 before heading to Southampton in 2012.
He leaves the Baggies having made 82 appearances scoring 29 goals.
He joins Erik Pieters as new signings for the Clarets after the Dutch full-back arrived from Stoke for an undisclosed fee on Monday.
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