The 29-year-old returns to his hometown club after spells at Southampton and the Baggies.
Rodriguez came through the Clarets youth system and made 129 first-team appearances before joining Southampton in 2012.
He went on to win an England cap in 2013, but injuries derailed his career at the Saints and he joined West Brom for £12 million ($15 million) two years ago.
Rodriguez scored 33 goals in 84 appearances for West Brom, including 22 in the Championship last season, but could not lead Albion back to the top flight at the first time of asking as they lost in the playoff semi-finals to Aston Villa.