The Clarets have been handed an FA Cup trip to Premier League rivals Fulham in the fourth round of the competition.
And the winners of the last-32 tie will then face a home game against either AFC Bournemouth or Crawley Town in round five.
Both rounds were drawn on Monday evening to aid fixture planning, with Burnley set to go to Craven Cottage on the weekend beginning Saturday, 23 January.
The tie will be a repeat of the 1962 FA Cup semi-final, which the Clarets won after a replay, with Burnley also winning the only other post-War FA Cup tie between the sides.
The Clarets drew 1-1 at Fulham in round five in February, 2003 before winning the replay 3-0 at Turf Moor ahead of a quarter-final defeat at
Fourth round tie gets BT Sport slot
Watford.
Should the Clarets progress against Fulham – who won 2-0 at QPR in the third round – they will play in round five at Turf Moor in the mid-week around Wednesday, 10 February.
The Clarets last faced Championship side Bournemouth, managed by ex-Burnley number two Jason Tindall, in the FA Cup in January, 2005, advancing with a 2-0 win.
However, should League Two side Crawley Town follow up their giant-killing win over Leeds by beating Bournemouth – and the Clarets get past Fulham – there will be history made at Turf Moor with a first ever meeting between the two clubs.
Full draws
Fourth round:
Cheltenham Town v Manchester City
Bournemouth v Crawley Town
Swansea City v Nottingham Forest
Manchester United v Liverpool
Southampton/Shrewsbury v Arsenal
Barnsley v Norwich
Chorley v Wolves
Millwall v Bristol City
Brighton v Blackpool
Wycombe v Tottenham
Fulham v Burnley
Sheffield United v Plymouth
Chelsea v Luton
Stockport/West Ham v Doncaster
Brentford v Leicester
Everton v Sheffield Wednesday
Fifth round:
Fulham or Burnley v AFC Bournemouth or Crawley Town
Manchester United or Liverpool v Stockport County or West Ham United or Doncaster Rovers
Sheffield United or Plymouth Argyle v Millwall or Bristol City
Chorley or Wolverhampton Wanderers v Southampton or Shrewsbury Town or Arsenal
Barnsley or Norwich City v Chelsea or Luton Town
Everton or Sheffield Wednesday v Wycombe Wanderers or Tottenham Hotspur
Swansea City or Nottingham Forest v Cheltenham Town or Manchester City
Brentford or Leicester City v Brighton & Hove Albion or Blackpool