The pressure, therefore, shifts onto the hosts this weekend, Manchester City, who will be expected to defeat the Swans but whose manager has also been the subject of scrutiny in recent weeks, especially following a 2-0 loss to Stoke City last weekend.
The Citizens aren't in the best form, sparking rumours of Pep Guardiola being ready to take Manuel Pellegrini's job, but will remain the favourites heading into this game as they aim to keep themselves firmly in the hunt in what is turning into a very open title race.
Two teams in poor form
Swansea come into this game with just one win from their last 11, their latest result a 3-0 defeat at the hands of league leaders Leicester City.
The Foxes ran riot at the Liberty Stadium, Riyad Mahrez bagging a hat-trick to down the Welsh side to 15th place, just two points above the drop zone.
The fear of a relegation scrap is now very much real, with pundits starting to be proven right as the positives begin to wear thin in South Wales - which was boldly underlined by Monk's sacking.
However, Pellegrini is also under pressure in Lancashire, his side boasting just three wins from their last seven games.
Comfortable wins against Southampton in the league and Hull City in the League Cup had many thinking they had bounced back, but a 2-0 loss at Stoke last time out, where they were completely out-played, has seen them come under fire once more.
Thus, they will be keen to bounce back on Saturday, and most will expect them to do so.
Stats in the Citizens favour
As well as overturning recently poor form, the visitors will also be looking to defy history's expectations, with them having won just three of the 12 meetings between these two clubs.
Two of those victories came all the way back in the early 1980s when these two first clashed in the top flight, with the only other being in 2012 when Luke Moore scored an 83rd minute winner, making up for Scott Sinclair, who would later go on to turn out for the Citizens, missing a penalty early on.
Since then, the Swans have taken just one of a possible 18 points off the Manchester club, that coming in a goalless draw at the Liberty in 2013.
If they do manage to defy the odds and get a victory on Saturday, it would not only be their first ever win at the Etihad, but the first time they have taken points off of the Sky Blues on the road in their entire history.
Swans can be boosted by team news
Nonetheless, though many things are not, the team news is certainly in Swansea's favour, with them boasting a fully fit squad whilst the Sky Blues are again without a handful of key players.
Their big absentees include Vincent Kompany, Samir Nasri, Pablo Zabaleta and Sergio Agüero. All are sidelined through injury and this has been felt by the side in recent weeks given their poor form.
Former-Swan Wilfried Bony could start once again for the hosts, but he has struggled to replicate his form with the Welsh outfit at the Etihad, scoring just eight goals in his 28 appearances so far, having managed 34 in 70 games for his previous employers and his opponents this weekend.
The Ivorian's more laid-back style is slammed as lazy by many, with him lacking the energy of Agüero, not to mention the composure. Swansea will hope he continues to justify these criticisms by lacking that cutting edge when they come up against him on Saturday.
Furthermore, after Raheem Sterling bagged a brace in a false nine role in midweek, his two goals turning the game against Borussia Mönchengladbach completely on its head, one would expect him to play there once more, even though Bony bagged a late goal himself off the bench.
The visitors, on the other hand, have a full strength squad available, bar the absence of Franck Tabanou - the back-up right-back injured, but yet to make his debut for the club anyway.
Curtis is likely to deploy a very different team to one that Monk might too, especially following another defeat, one labelled as particularly poor this time as well in contrast to the positives that could be taken from a loss at Anfield.
Mo Barrow and Jefferson Montero will hope that they have put themselves in contention for a starting berth after both made an impact off the bench in that Leicester defeat, whilst many are calling for the return of Jack Cork, the Englishman able to help his team retain possession and actually do something with the ball.
Another popular opinion is for Kyle Bartley to be dropped after another terrible performance in the centre of defence, with Federico Fernandez hoping for a recall.
Jonjo Shelvey will also hope to come back into the team after missing out on the match day squad completely on his return from suspension last week,.
Given recent results, it seems that big changes like these will be needed for Swansea to improve.
Manchester City will host the Welsh outfit on Saturday afternoon at the Etihad Stadium, with kick-off at 3pm.