The Reds will host Marco Silva's strugglers at Anfield on Wednesday night in the Premier League before the Toffees cross Stanley Park again on the first weekend of January for a knockout clash.
It will be the rivals' 25th meeting in the FA Cup and the 286th meeting of the two teams since their first one in 1894.
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Arsenal, who have lifted the FA Cup 13 times, were drawn against Championship side Leeds, whose only FA Cup win came against the Gunners in the 1972 final.
Holders Manchester City host League Two side Port Vale, while Manchester United visit Wolverhampton Wanderers.
AFC Fylde of the National League, the lowest-ranked side guaranteed a third round spot, were drawn away to Premier League Sheffield United.
Boston United of the National League North, the sixth tier of English football, could face Newcastle United if they overcome Rochdale in a second-round home replay.
Chelsea host Nottingham Forest, Tottenham Hotspur travel to Middlesbrough and Leicester City welcome Wigan Athletic in the pick of the other games.
The ties will between January 3 and January 6.
FA Cup third round draw in full
Leicester City v Wigan Athletic
QPR v Swansea City
Fulham v Aston Villa
Chelsea v Nottingham Forest
Wolves v Manchester United
Charlton Athletic v West Brom
Rochdale or Boston United v Newcastle United
Cardiff City v Forest Green Rovers or Carlisle United
Oxford United v Exeter City or Hartlepool United
Sheffield United v AFC Fylde
Southampton v Huddersfield Town
Liverpool v Everton
Bristol City v Shrewsbury Town
Bournemouth v Luton Town
Brighton v Sheffield Wednesday
Bristol Rovers or Plymouth Argyle v Coventry City or Ipswich Town
Eastleigh or Crewe Alexandra v Barnsley
Manchester City v Port Vale
Middlesbrough v Tottenham
Reading v Blackpool
Watford v Tranmere Rovers
Preston v Norwich City
Millwall v Newport County
Crystal Palace v Derby County
Solihull Moors or Rotherham United v Hull City
Brentford v Stoke City
Fleetwood Town v Portsmouth
Arsenal v Leeds United
Gillingham v West Ham United
Burton Albion v Northampton Town
Burnley v Peterborough United
Birmingham City v Blackburn Rovers