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.
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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
