United boss interim boss Ralf Rangnick's first FA Cup tie will be an all-Premier League encounter with Villa at Old Trafford, where former Liverpool captain Gerrard can expect a hostile reception.
Holder Leicester will again come up against their former manager Claudio Ranieri, who masterminded the club's fairytale Premier League triumph in 2016, at the King Power Stadium.
The Foxes lifted the trophy for the first time in their history in May after beating Chelsea 1-0 in the final at Wembley.
Chelsea's bid to go one better than last season commences when they welcome National League leaders Chesterfield to Stamford Bridge.
Liverpool and Tottenham host League One opposition in the form Shrewsbury Town and Morecambe respectively, while Premier League leaders Manchester City travel to League Two Swindon Town.
Record 14-time winners Arsenal head to Nottingham Forest aiming to avoid a repeat of their 2018 meeting at this stage of the competition, which Forest won 4-2.
All ties will be played during between January 7-10.
FA Cup third round draw in full:
Boreham Wood or St Albans City v AFC Wimbledon
Yeovil Town v AFC Bournemouth
Stoke City v Leyton Orient
Swansea City v Southampton
Chelsea v Chesterfield
Liverpool v Shrewsbury Town
Cardiff City v Preston North End
Coventry City v Derby County
Burnley v Huddersfield Town
West Bromwich Albion v Brighton & Hove Albion
Kidderminster Harriers v Reading
Leicester City v Watford
Mansfield Town v Middlesbrough
Hartlepool United v Blackpool
Hull City v Everton
Bristol City v Fulham
Tottenham v Morecambe
Millwall v Crystal Palace
Port Vale v Brentford
Swindon Town v Manchester City
Wigan Athletic v Blackburn Rovers
Luton Town v Harrogate Town
Birmingham City v Plymouth Argyle
Manchester United v Aston Villa
Wolverhampton Wanderers v Sheffield United
Newcastle United v Cambridge United
Barnsley v Ipswich Town or Barrow
Peterborough United v Bristol Rovers
West Ham United v Leeds United
QPR v Rotherham United
Charlton Athletic v Norwich City
Nottingham Forest v Arsenal