The ties will be played in the week commencing August 23.
Barrow beat Scunthorpe in the first round on Tuesday night and their tie against Villa will be played in the week commencing August 23.
Elsewhere in Wednesday’s draw, which was separated into northern and southern sections, Newcastle will host fellow Premier League side Burnley while Watford were drawn at home to Crystal Palace.
Newport’s reward for knocking out Ipswich is a home tie against Southampton, while Everton travel to Sky Bet Championship side Huddersfield.
Leeds host League One Crewe, who edged past Hartlepool in the first round despite being down to 10 men after only 13 minutes, while Arsenal are away at West Brom.
Forest Green travel to Premier League new boys Brentford, while Norwich, also back in the top flight, face Bournemouth, who failed to secure an instant return.
Gillingham beat Crawley 10-9 on penalties and will host Cheltenham, and Wolves travel to Nottingham Forest while Brighton will travel to Cardiff.
Carabao Cup second round draw
Northern Section: Oldham v Accrington, Newcastle v Burnley, Wigan v Bolton, Huddersfield v Everton, Sheffield United v Derby, Stoke v Doncaster, Shrewsbury v Rochdale, Nottingham Forest v Wolves, Morecambe v Preston, Blackpool v Sunderland, Leeds v Crewe, Barrow v Aston Villa.
Southern Section: Brentford v Forest Green, Millwall v Cambridge, West Brom v Arsenal, Norwich v Bournemouth, Cardiff v Brighton, Birmingham v Fulham, Gillingham v Cheltenham, QPR v Oxford, Swansea v Plymouth, Stevenage v Wycombe, Newport v Southampton, Northampton v AFC Wimbledon, Watford v Crystal Palace.