The Toffees face Jurgen Klopp's side on Thursday (AEDT) in the Premier League and will make the short trip to Anfield for a second time this season during the first week of January.
Pep Guardiola's City thumped Watford 6-0 in last season's final and have been given a favourable draw to kick-start their 2019-20 campaign, with John Askey's League Two side pitching up at the Etihad Stadium.
Manchester United faces a difficult away tie against Premier League rivals Wolves, who knocked them out of the competition last season's quarter-finals, while Arsenal welcome Championship leaders Leeds United to the Emirates Stadium.
New Tottenham boss Jose Mourinho takes his side to Middlesbrough, managed by former Spurs defender Jonathan Woodgate, and Chelsea host Nottingham Forest.
Ties will take place between January 3 and 6.