Both bosses appear to be losing the patience of their fanbases following a run of disappointing form, with the Gunners at risk of missing out on the top four and United being drawn into the relegation mire.
After 21 years of loyal service, could these be the final weeks of Wenger's long-running reign in North London? Comments made by chief executive Ivan Gazidis at the weekend would certainly suggest so, though Arsenal tend to operate differently to every other club in the division.
On the back of his worst set of results in two long decades at the helm, Wenger has attempted to bat away the constant questions regarding his future by insisting that an official announcement will be made "soon".
That was the stance taken following the 3-1 loss to West Bromwich Albion a fortnight ago - arguably the nadir in this season that has shown little signs of improvement - and the Frenchman again held firm when questioned in the wake of Sunday's 2-2 draw with Manchester City.
The character shown against the Citizens did at least offer a response to some of Wenger's critics, with Arsenal twice battling from behind to earn a point against one of their top-four rivals, yet it says an awful lot about their current footing that simply not losing is seen as a big positive.
Rather than closing the gap on fourth-placed City they still remain seven points adrift, with a game in hand to play, while Manchester United in fifth are two points better off heading into the final run-in.
The one constant that Wenger has had to fall back on in recent years, finishing in the top four and therefore securing Champions League football, is no longer a given, meaning that winning the FA Cup for a third time in four years is now his only real hope of appeasing disgruntled supporters.
Fights broke out around the Emirates Stadium during the draw with City at the weekend, with Wenger himself admitting that the angst from the stands was being felt by his players on the field, having seen his side win just three of their last 10 games in all competitions - two of those coming against non-league opposition in the FA Cup.
While the players wait for an announcement on their manager's future, and the uncertainty stretches from days to weeks to months, all they can do is attempt to get their campaign back on track and potentially close the gap on the teams immediately above them.
For the first time since 1995 local rivals Tottenham Hotspur will almost certainly finish as top dogs in North London, barring the most incredible of collapses, but it is the failure to secure top European football that may well prove to be the final nail in the coffin for Wenger. Defeat to the Hammers on Wednesday and the Wenger Out brigade may well get their wish.
Recent form: WLLWLD
Recent form in Premier League: LLWLLD
West Ham United
Wenger is not the only Premier League manager under pressure right now and, if the bookies are to be believed, it is opposite number Bilic - along with Sunderland's David Moyes - who is the most likely to leave his post next.
The vote of confidence given to him by the club's board on Sunday counts for little in this day and age, coming two months on from Claudio Ranieri also being given the full backing of his peers - 16 days later the Italian was out of a job.
Bilic may well point his finger at the board themselves should he indeed be axed in the coming weeks, having been promised big names last summer only to settle for, to put it kindly, unwanted personnel from elsewhere across Europe.
Losing Dimitri Payet in January certainly did not help matters, either, and since then the Hammers have won just one of their eight matches, losing four on the bounce to find themselves back in the relegation mix once again.
That has left West Ham joint-bottom of the form table, but ahead of their trip to the Emirates Stadium they can perhaps take solace from the fact that they have already snatched points off Liverpool and Manchester United on their travels this season, while coming within a minute of doing so against Tottenham.
United have scored in all but one of their last 10 matches, though at the other end of the field they are unable to keep the opposition out and now hold claim to the third-worst defensive record in the entire division, shipping 13 in their last six matches.
Six points now stand between West Ham and the dropzone, putting added significance on the upcoming meetings with strugglers Swansea City and Sunderland in the next fortnight - lose those and the board may well be backtracking on the statement released at the weekend.
One man who could prove vital in Wednesday's London derby is Andy Carroll, who brought up his 50th Premier League goal with the opener in the 2-1 loss to Hull City - five of those previous goals coming against Arsenal, the team he likes playing against more than any other opponent.
Recent form in Premier League: DDLLLL
Danny Welbeck has managed to get a run of starts under his belt in the last month, but it remains to be seen whether he will be risked from the off twice in the space of three days, with Olivier Giroud the leading candidate to start up top.
There are doubts over Aaron Cresswell, too, having come off injured at the KCOM Stadium, and in the absence of Winston Reid and Angelo Ogbonna there is expected to be another start handed to experienced centre-back James Collins.
Arsenal possible starting lineup:
Ospina; Bellerin, Mustafi, Paulista, Monreal; Xhaka, Coquelin; Walcott, Ozil, Sanchez; Giroud
West Ham United possible starting lineup:
Randolph; Byram, Collins, Fonte, Masuaku; Noble, Kouyate; Antonio, Lanzini, Snodgrass; Carroll
Head To Head
Arsenal have tasted defeat in just one of their last 17 Premier League meetings with West Ham, winning 13 and drawing three of those, though that sole loss did come in this fixture last season.
United have five away wins against Arsenal in the Premier League, while the Gunners are looking to pull off the double over their London rivals for the 10th time in the competition.
We say: Arsenal 2-1 West Ham United
Even taking the recent draw with Man City into consideration, Arsenal are very much a team out of form right now. To make matters worse for the Gunners, they have won just one of their last eight midweek league matches, but they should have enough to scrape past an equally out of sorts Hammers side on their own patch.