The Hammers' boss is the current bookies' favourite to be the next Premier League manager to leave, after a humbling defeat at the City Ground last week.
Match preview
After some positive results at home, all that good work was undone for West Ham as they fell to a 3-0 defeat away to Nottingham Forest last time out, heaping the pressure on manager Lopetegui, and leaving some supporters pining for the return of David Moyes, who was hounded out of the club at the end of last season.
Considering the club spent multi-millions of pounds over the summer to back the new manager, they are down in 14th and just six points above the relegation zone.
Two wins in their last nine games across all competitions is a woeful return, but they did both come in their last two matches here at the London Stadium.
A 4-1 win over Ipswich Town before the last international break was followed by a 2-1 success over Manchester United, which resulted in Erik ten Hag's sacking.
So despite all of the negative atmosphere around the club, this fixture presents West Ham with the chance to win three straight home games in the league in the same campaign for the first time in over two years.
The attacking woes of their opponents Everton will also give them belief of a rare clean sheet on home soil, as West Ham have not kept one here in 14 games - the worst ongoing run of any team in the Premier League, and the club's worst-ever run in the competition.
Not since the mid-1960s have West Ham gone so long without a home clean sheet, and a recent upturn in results against Everton suggests they could end that run.
The Toffees used to be something of a bogey team for West Ham, but Moyes was often the scourge of his former employers, contributing to a run of five wins in the last eight head-to-heads - as many wins as West Ham managed in the 30 meetings prior.
After their own humbling defeat at bottom-of-the-table Southampton last week, patience is starting to run thin with Dyche among the Everton fanbase.
There is huge respect for what he has done in trying circumstances over the past two years, but fans are starting to turn in large numbers due to the sluggish, defensive-first football that is often on show.
Everton were very lucky to snatch a point against Fulham two weeks ago, and while they, and more specifically Beto, suffered from misfortune last weekend at St Mary's, they did not do enough to win before falling to a late defeat.
That result has birthed a rather damning statistic for the club, as they have now lost four of the last five games against teams starting the day bottom of the Premier League table.
Along with Southampton, the rest of the bottom four all picked up points last weekend, pushing Everton slightly closer to the drop zone again, but they will at least avoid that fate heading into the international break regardless of what happens here, with the gap to Ipswich currently four points.
Despite winning on this ground last season, results on the road in general and against West Ham of late offer little for Everton fans to get excited about though.
The win at Portman Road last month remains their only one away from home in the league in 2024, so if Everton are to add to that, they will hope they can rediscover their good previous form against the Hammers, the team they have beaten most often in the Premier League - 29 times.
Suffering away supporters will know how arduous trips to London often are for the Toffees though, and they have lost their last three in capital conceding 12 in the process to Chelsea, Arsenal and Tottenham Hotspur, but West Ham proven kinder opponents in the past.
Almost one-third of Everton's 40 Premier League wins in the capital have come against West Ham, and that tally of 13 also represents the highest number of away wins they have enjoyed against a single club in the competition.
West Ham United Premier League form:
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West Ham United form (all competitions):
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Everton Premier League form:
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Team News
Mohammed Kudus will be a big miss for the home side, after the Ghanaian had his ban extended to five matches following the outburst against Spurs which resulted in a sending off for violent conduct.
Lopetegui will also be without Edson Alvarez, who was dismissed at Forest last week for two quick-fire bookings, meaning Guido Rodriguez will likely escape being dropped due to a lack of options in the middle, after he was hooked at half time last week.
Niclas Fullkrug will remain out for a few more weeks at least, and Lopetegui confirmed there are "little issues" in his squad at present, possibly referring to Tomas Soucek and goalkeeper Alphonse Areola, who both missed out entirety last week through illness and injury, respectively.
After coming under fire for leaving Jarrad Branthwaite on the bench in the last two games, Dyche will most likely bring the highly-sought-after defender back into the side here, but whether it is for Michael Keane or James Tarkowski is not clear, given the concerning form of the latter.
Seamus Coleman is the only new injury concern for Everton, after sustaining a hamstring problem this week, while Abdoulaye Doucoure could return sooner than expected after missing the defeat at Southampton.
The shouts are getting louder for Dyche to hand Beto an opportunity from the start up front after two promising cameos from the bench, paired with a dismal run of form for Dominic Calvert-Lewin, but after the manager stuck with the first-choice number nine during a torrid scoring drought last season, dropping him appears unlikely.
West Ham United possible starting lineup:
Fabianski; Wan-Bissaka, Mavropanos, Todibo, Kilman, Emerson; Rodriguez, Carlos Soler; Bowen, Ings, Paqueta
Everton possible starting lineup:
Pickford; Young, Tarkowski, Branthwaite, Mykolenko; Gueye, Doucoure; Lindstrom, McNeil, Ndiaye; Calvert-Lewin
We say: West Ham United 2-1 Everton
West Ham's only positive results of late have been in the league at home, while Everton are notoriously among the worst travellers in the Premier League.