The Whites met The Robins in the same round of last season's competition, winning on penalties after the game finished 2-2 in normal time with former keeper Paul Robinson scoring in the final minute to take the tie to extra-time.
Swindon took around 2,500 supporters last time and another big gate will be expected after over 30,000 watched the Huddersfield game in the first round last night.
The match is due to be played at Elland Road during the week commencing September 20.
The second round draw in full:
Burnley v Wolves
Coventry v Sheffield Wednesday
MK Dons v Cardiff City
Leicester v Mansfield/Preston
Oldham v Spurs
Colchester v WBA
Reading v Watford
Wrexham v Sheffield Utd
Nottingham Forest v Rotherham
Norwich City v Bristol Rovers
Crewe v Sunderland
Tranmere v Portsmouth
Manchester City v Barnsley
Birmingham v Lincoln City
Northampton Town v Southampton
Blackburn v Bournemouth
Grimsby v Charlton
Crystal Palace v Hartlepool
Boston/Luton v Fulham
West Ham v Notts County
Aston Villa v QPR
Doncaster v Ipswich
Yeovil v Bolton
Bristol City v Everton
Leeds v Swindon