The Blues played the Clarets at the Etihad in October and will host Sean Dyche’s men again in early January.
The tie is provisionally scheduled to take place on 6 January, 3pm but TV schedules could possibly mean this date or time could change.
Ticket details will be released in due course.
The teams last met in the FA Cup in January 1991, with Colin Hendry’s goal giving the Blues a 1-0 win at Turf Moor.
Prior to that, the teams hadn’t met since 1933 with Fred Tilson again giving the Blues a 1-0 win at Turf Moor.
The very first - and only other meeting - was also at Burnley in 1931, but on that occasion the hosts won 3-0.
FA Cup Third Round draw in full:
Ipswich v Sheffield United
Watford v Bristol City
Birmingham v Burton
Liverpool v Everton
Brighton v Crystal Palace
Aston Villa v Woking or Peterborough
Bournemouth v AFC Fylde or Wigan
Coventry v Stoke
Newport v Leeds
Bolton v Huddersfield
Port Vale or Yeovil v Bradford
Nottingham Forest v Arsenal
Brentford v Notts County
QPR v MK Dons
Manchester United v Derby
Doncaster v Slough Town or Rochdale
Tottenham v AFC Wimbledon
Middlesbrough v Sunderland
Fleetwood or Hereford v Leicester
Blackburn or Crewe v Hull
Cardiff v Mansfield
Manchester City v Burnley
Shrewsbury v West Ham
Wolves v Swansea
Stevenage v Reading
Newcastle v Luton
Millwall v Barnsley
Fulham v Southampton
Wycombe v Preston
Norwich v Chelsea
Gillingham or Carlisle v Sheffield Wednesday
Ties to be played January 5-7
