The 22-year-old has not played a first-team game for Chelsea in his four and a half years with the club following his move from Nottingham Forest.
Bamford has had loan spells with MK Dons, Derby, Middlesbrough, Crystal Palace and Norwich.
Speaking after Saturday's 3-0 Premier League defeat at Stamford Bridge, Clarets boss Sean Dyche had said the signing was imminent.
Burnley have signed Chelsea striker Patrick Bamford on a season-long loan.