The tie will be played at The Jonny-Rocks Stadium (the best named ground in England?) at a date yet to be confirmed.
City and Cheltenham Town have never met competitively before.
The Robins are fifth in League Two, four points off the top.
Michael Duff's side have reached this stage after beating South Shields, Crewe Alexandra and Mansfield to set up the fourth round tie.
This is only the third time Cheltenham have reached this stage of the competition.
Full 4th round draw:
Cheltenham v Manchester City
Bournemouth v Crawley
Swansea v Nottingham Forest
Manchester United v Liverpool
Southampton or 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 or West Ham v Doncaster
Brentford v Leicester
Everton v Sheffield Wednesday
The ties will be played over the weekend of 22-25th January.
The FA Cup 5th round draw was also made this evening.
If City beat Cheltenham Town, we will face either Swansea City or Nottingham Forest at The Liberty Stadium or City Ground.
The full 5th round draw is:
Fulham or Burnley v Bournemouth or Crawley
Manchester United or Liverpool v Stockport or West Ham or Doncaster
Sheffield United or Plymouth v Millwall or Bristol City
Chorley or Wolves v Southampton or Shrewsbury or Arsenal
Barnsley or Norwich v Chelsea or Luton
Everton or Sheffield Wednesday v Wycombe or Tottenham
Swansea or Nottingham Forest v Cheltenham or Manchester City
Brentford or Leicester v Brighton or Blackpool
Ties to be played February 9-11.