City will welcome their arch-rivals to the Etihad Stadium still revelling in their 6-1 league triumph at Old Trafford in October.
The tie was one of only two all-Premier League affairs to emerge from Sunday afternoon's draw, with Newcastle hosting Blackburn in the other.
Manager Paul Buckle described himself as "delighted" after Bristol Rovers landed a home draw against Aston Villa.
Rovers won 6-1 at Totton on Sunday to book their meeting with Alex McLeish's side on the weekend of January 7-8.
Buckle told ITV1: "I am delighted, absolutely thrilled.
"I am delighted for the fans and players. We've got a lot of tough league fixtures in between.
"We'll look forward to it but there is a lot to come before then."
Tamworth, of the Blue Square Premier, will travel to Everton and manager Marcus Law admitted it will be a dream come true facing a top-flight team.
He told Sky Sport News: "I'll be working and planning very hard come that time."
Goalkeeper Jonathan Hedge added: "It's what dreams are made of, we'll go there full of confidence.
"I'm a Sheffield United fan so I would have liked them or one of the two Manchester clubs but we will have to make do with Everton."
Norwich will take on Championship side Burnley at Carrow Road and manager Paul Lambert expects the Clarets to put up a tough challenge.
He told www.canaries.co.uk: "Eddie Howe has stabilised things there and they are going well.
"They got a great win at West Ham on Saturday which is certainly no mean feat and just shows Burnley are tough opposition.
"So it will be a tough game but at least we're at home which is the first big thing you can hope for.
"I'm sure we'll get another good crowd at Carrow Road and it will be another big occasion for our supporters.
"I've said before that we go into every game and every competition looking to win and do as well as we possibly can.
"We will have to be on our toes against Burnley that's for sure but it's another game we'll be looking forward to."
The draw for the third round of the FA Cup is as follows:
Middlesbrough v Shrewsbury
Nottingham Forest v Leicester
Manchester City v Manchester United
Dagenham & Redbridge/Walsall v Millwall
Crawley v Bristol City
Doncaster v Sutton United/Notts County
Bristol Rovers v Aston Villa
Tottenham v Cheltenham
Sheffield Wednesday v West Ham
MK Dons v QPR
Hull v Ipswich
Coventry v Southampton
Brighton v Wrexham
Fulham v Charlton
Birmingham v Wolves
Norwich v Burnley
Arsenal v Leeds
Derby v Crystal Palace
Fleetwood/Yeovil v Blackpool
Swindon v Wigan
Barnsley v Swansea
Chelmsford/Macclesfield v Bolton
Newcastle v Blackburn
Everton v Tamworth
Sheffield United v Salisbury/Grimsby
Liverpool v Southend/Oldham
Gillingham v Stoke
Chelsea v Portsmouth
Watford v Bradford
Peterborough v Sunderland
West Brom v Cardiff
Reading v Stevenage
