The Citizens have won the competition on three previous occasions, while in contrast the hosting Merseysiders have never lifted the trophy.
The Goodison Park faithful may not quite be in mutinous mood, but there is a feeling that the natives are becoming increasingly restless with what is being produced on the pitch.
On paper at least manager Roberto Martinez has at his disposal one of the strongest Everton squads in recent memory, yet results in recent weeks have failed to live up to that fact.
Of their 11 home matches in the Premier League this season, the Toffees have been triumphant just three times. Last time out Martinez's charges played out a 1-1 draw with Tottenham Hotspur, although it was a game that was largely dominated by the hosts, who saw efforts strike the crossbar and the upright.
Nevertheless, Martinez used his pre-match press conference to urge supporters to stay behind the team when he said: "We know that we showed great signs this season but we need to be patient. The performances have been very inconsistent and probably that's where the frustration comes. We all need to learn as a football club that we've got a special group of players and we have to help them."
On a more positive note, Everton do boast one of the country's in-form strikers. Romelu Lukaku has been involved in 10 goals in his last six Premier League outings at Goodison Park (seven goals and three assists).
The League Cup is a trophy that has so far escaped Everton's clutches, with their two appearances in finals - against Aston Villa in 1977 and Liverpool seven years later - ending in defeat after replays.
Recent form: DDLWLD
As far as the bookmakers are concerned, the Etihad Stadium is Guardiola's preferred destination - the thinking being that the former Barcelona and current Bayern Munich head coach will be able to catapult City into the higher echelons of European football in a way that has so far eluded Pellegrini.
From the point of view of the City boss, he will at least be able to depart the Mancunians with his pride intact, providing that he is able to guide the team to a couple of trophies between now and the season's conclusion.
The League Cup is a competition that the Chilean is bound to be fond of, given that it provided him with his first taste of glory on English soil in March 2014 when City overturned a one-goal deficit to beat Sunderland 3-1.
If City are to take a positive result with them back to the Etihad for the second leg later this month, they will need to arrest some poor away form of late. The 2-1 victory at Watford on Saturday night was their first on the road in the Premier League in seven attempts.
As well as the success of two years ago, City won the tournament in 1970 and 1976 courtesy of 2-1 wins over West Bromwich Albion and Newcastle United respectively.
Recent form: WWLWDW
Pellegrini confirmed to reporters during his press conference that Wilfried Bony is not fit enough to be included in the City squad for this clash, which should see Sergio Aguero spearhead the attack.
The visitors will still be without their captain Vincent Kompany, though, as the Belgian centre-back continues to recover from a calf problem.
Samir Nasri is the only other absentee that Pellegrini will have to contend with when it comes to team selection.
Everton possible starting lineup:
Robles; Coleman, Stones, Funes Mori, Baines; Cleverley, Barry; Deulofeu, Barkley, Kone; Lukaku
Man City possible starting lineup:
Hart; Sagna, Otamendi, Mangala, Kolarov; Toure, Fernandinho, De Bruyne, Silva, Sterling; Aguero
Head To Head
In terms of the League Cup, the two sides have crossed paths twice.
We say: Everton 1-2 Man City
Everton tend to give City a tough game at Goodison and that trend is likely to continue. But, with Aguero looking sharper by the game and the likes of Kevin De Bruyne, Raheem Sterling and David Silva behind the Argentine, the away side could well have too much.