Last season, the 29-year-old winger joined Turkish giants Besiktas on a free transfer and made 25 league appearances, scoring five goals and assisting five goals.
Redmond’s contract with the Turkish outfit expired at the end of June and the player is now a free agent.
Vincent Kompany wants a right-winger this summer and Southampton’s Nathan Tella is his priority candidate.
But reluctant to meet Southampton’s high transfer fee demands, Burnley have shifted their focus to Redmond.
Redmond, who has played over 250 games in the Premier League, is in negotiations with the Turf Moor outfit.
Now, according to Turkish outlet Sports Digitale, Burnley have reached an agreement with Redmond to sign him this summer.
The news will be a blow to Besiktas, who had been hoping to keep hold of Redmond following his displays last season.
But it appears the winger is poised to return to England.
