Newcastle have eliminated Manchester City and Manchester United in the past two rounds and travel to Stamford Bridge for the next round.
Everton entertain Fulham in an all-Premier League tie and League One outfit Port Vale play at home against Middlesbrough of the Championship.
Matches will be played in the week commencing 18 December, 2023.
On Tuesday, Port Vale moved into the quarter-finals for the first time in their history with a 1-0 win at League Two Mansfield Town, while Middlesbrough came from behind to defeat Exeter City 3-2.
Newcastle beat last season's winners Manchester United 3-0 in a rematch of the 2022-23 final, with West Ham also scoring three times in their 3-1 victory against an Arsenal side that are second in the Premier League.