The 27-year-old turned down a new deal at Ashton Gate after his contract ran out following the club's relegation and undertook a medical at Burnley's Gawthorpe training headquarters on Friday.
With a deal expected to be confirmed on Monday, the former Manchester United shot-stopper will draw a close to his time with the Robins having made 27 appearances since signing a one-year contract in July 2012.
Sean Dyche made his move quickly after the Turf Moor club were also left short on goalkeeping options following the departure of Lee Grant and long-serving Dane, Brian Jensen.
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