The Blues enjoyed a 3-0 home victory over Salford City in the second round on Wednesday, and will travel to Joey Barton’s Fleetwood on September 23 in their next step on the road to Wembley.
Should they see off the League One side, they will welcome the winner of the third round tie between the Hammers and the Tigers.
The fourth round ties will be played one week after the third round, during the week commencing September 28.
Both West Ham and Hull suffered defeat in their last trips to Goodison Park; the Hammers lost 2-0 there last October, and the Tigers were hammered 4-0 by Ronald Koeman’s Everton under Marco Silva.
West Ham beat Charlton to make the third round, while Hull have so far knocked out Sunderland and Leeds, both on penalties.
Everton have never won this competition, and lost on penalties by Leicester City in the quarter-finals last season.
Should the Blues make the fourth round, it will mean they will have played six matches in the space of just two-and-a-half weeks by the end of September.
Everton - potential September fixture list
Sunday, September 13 - Tottenham 0-1 Everton (Premier League)
Wednesday, September 16 - Everton 3-0 Salford (EFL Cup second round)
Saturday, September 19 - Everton vs West Brom (Premier League, 12.30pm BST)
Wednesday, September 23 - Fleetwood vs Everton (EFL Cup third round, 7.45pm BST)
Saturday, September 26 - Crystal Palace vs Everton (Premier League, 3pm BST)
Week commencing September 28 - Fleetwood or Everton vs West Ham or Hull (EFL Cup fourth round)
Here’s the full Fourth Round draw.