The Toffees overcame Salford City and Fleetwood Town to advance to the last 16, whereas West Ham knocked out Charlton Athletic and Hull City to set up a showdown with the Toffees.
There is certainly an air of optimism around Goodison Park at the moment, with Everton having enjoyed their best start to a league season since the 1993-94 campaign.
Carlo Ancelotti's men maintained their 100% start to the new campaign at the weekend as they overcame Crystal Palace 2-1 at Selhurst Park, but the story could have panned out differently had it not been for a controversially-awarded penalty for handball against Joel Ward.
New signings Allan, Abdoulaye Doucoure and James Rodriguez have slotted right into the Toffees team, while Dominic Calvert-Lewin has enjoyed a particularly fruitful start to the campaign with five goals thus far, and a strong run in the EFL Cup would have the Everton faithful dreaming of further success under the tutelage of Ancelotti.
The Toffees overcame Salford and Fleetwood by a combined scoreline of 8-2 to reach the fourth round of the EFL Cup, and while Everton have reached the final twice in 1977 and 1984, they are yet to lift the trophy.
Should Everton book their spot in the fifth round this week, it will mark only the fourth time since the 2010-11 season that the Toffees have advanced further than the fourth round stage, with Leicester City knocking Duncan Ferguson's side out on penalties in last season's quarter-finals.
Meanwhile, West Ham's third-round drubbing of Hull was marred by a trio of positive coronavirus tests just before the game began, with manager David Moyes as well as Josh Cullen and Issa Diop being sent home from the London Stadium.
In spite of that health scare, the Hammers prevailed in convincing fashion against their League One opponents to advance to the fourth round in their quest for a first-ever EFL Cup triumph.
Moyes's side seem to be turning their fortunes around in the Premier League since that dismal opening-day defeat to Newcastle United - West Ham were unlucky not to get something from their game at Arsenal before trouncing Wolverhampton Wanderers 4-0 at the weekend - with Alan Irvine on the touchline while Moyes took the reins from home.
As is the case with Everton, West Ham have reached the final twice but are yet to taste success in the competition, with the capital outfit losing the 1966 and 1981 finals to West Bromwich Albion and Liverpool respectively.
West Ham will no doubt hope to record a victory here to boost their confidence heading into a daunting run of October fixtures - the Hammers must take on Leicester City, Tottenham Hotspur, Manchester City and Liverpool all within the next five weeks.
Everton took the spoils in a 2-0 win when they welcomed West Ham to Goodison Park in the Premier League last season, and the two sides have not faced off in the EFL Cup since the 2007-08 campaign, where Moyes enjoyed a 2-1 quarter-final victory with the Toffees against his current club.
Everton EFL Cup form: WW
Everton form (all competitions): WWWWW
West Ham United EFL Cup form: WW
West Ham United form (all competitions): LWLWW
A cup run would do both of these clubs the world of good, so the two managers are expected to name relatively strong lineups for Wednesday's encounter.
Jordan Pickford has drawn criticism on several occasions this season and could see his spot for this game come under threat by Jonas Lossl, while Bernard and Gylfi Sigurdsson are among those in line to make the first XI.
Changes are expected up top with Sebastien Haller - who currently leads the way in the EFL top scorer charts with four strikes to his name - in line to spearhead the Hammers attack.
Everton possible starting lineup:
Pickford; Kenny, Mina, Keane, Digne; Davies, Sigurdsson, Doucoure; Iwobi, Calvert-Lewin, Bernard
West Ham United possible starting lineup:
Randolph; Johnson, Balbuena, Ogbonna, Masuaku; Noble, Rice; Yarmolenko, Lanzini, Anderson; Haller
We say: Everton 3-1 West Ham United
West Ham supporters will be hoping that their impressive victory over Wolves will be the one that turns the corner, but the Hammers are now facing a challenging cup fixture against one of the most in-form sides in the Premier League. Changes are expected in both camps but we cannot see Everton's strong run of form ending anytime soon, so we expect Ancelotti's men to prevail and book their spot in the quarter-finals.