Now, though, as they prepare to travel to Stamford Bridge on Saturday afternoon, they may find next term that the roles have been reversed.
The 2-0 defeat away at Everton last weekend ensured that it would be a season to forget for everybody connected with Chelsea.
Already out of the Champions League and massively adrift of the title race, the FA Cup represented the only chance of silverware for the West Londoners. Yet, it was a familiar face - Romelu Lukaku - that produced the goods for Everton as he dumped his former employers out of the competition at the quarter-final stage.
If nothing else, interim boss Guus Hiddink has at least steadied the ship since taking interim charge in December. When the Dutchman strode into Stamford Bridge for a second time, he inherited a team that was just one point above the bottom three.
But, it is worth noting that while Chelsea have lost none of the 13 Premier League matches since Jose Mourinho was sacked, fewer than half of those (six) have ended in victories.
Now, the focus will be on finishing as high up the Premier League as possible. They currently sit 10th, but if they can turn some of those draws into victories, seventh or maybe even sixth position will not be beyond them.
Fourth place is too much of a gap to bridge, though - 11 points to be exact. Nevertheless, the Blues can have a big say on who finishes in the Champions League places given that after West Ham they also face Manchester City, Tottenham Hotspur and Leicester City.
Recent form: DDWWWD
Recent form (all competitions): WWWDLL
At the start of the season, not even the most ardent of West Ham supporters would have put money on their side being just two points outside of the Champions League places with nine matches left to play.
After all, a manager was hired with no previous experience of coaching in the Premier League, while decent-sized fees were splurged on the likes of Dimitri Payet, Angelo Ogbonna, Pedro Obiang and Michail Antonio, none of whom had ever played in England's top flight.
An early exit from the Europa League, coupled with two home Premier League defeats at the start of the season, did little to deter the doubters. Yet, since then, just four losses have been suffered in the league, while entertaining football has also been thrown into the bargain.
While many players have contributed, it is hard to ignore the impact made by the classy Payet. The French playmaker has scored eight Premier League goals and assisted a further seven.
He has also proven to be the key man in West Ham's run to the quarter-finals of the FA Cup by scoring stunning free kicks against Blackburn Rovers and Manchester United.
Of course, it all means that those fans that make the short trip to Stamford Bridge on Saturday will be doing so in expectation rather than hope, when for so long it has been the other way round.
Recent form: WLDWWW
Recent form (all competitions): DWWWWD
West Ham centre-back Winston Reid limped off with what appeared to be a recurrence of a hamstring complaint last time out against Man United.
However, it has since been confirmed that it was only a precaution and the Kiwi will be fit to face Chelsea.
Aside from the absent duo, Bilic has a full complement of players to select from, with Sam Byram expected to be back in the squad after he missed the United game because he was cup-tied.
As for the home side, they will have to make do without Diego Costa, who is suspended as a result of the red card that he received at Everton.
Chelsea possible starting lineup:
Courtois; Azpilicueta, Cahill, Terry, Ivanovic; Mikel, Matic; Willian, Fabregas, Pedro; Traore
West Ham possible starting lineup:
Adrian; Antonio, Reid, Ogbonna, Cresswell; Kouyate, Noble, Lanzini, Payet; Emenike, Carroll
Head To Head
Since May 2003, this fixture has been won by West Ham on only two occasions.
As far as their last win at Stamford Bridge goes, you have to go back a little bit further to September 2002 when Paolo Di Canio notched a brace in a 3-2 triumph.
We say: Chelsea 0-0 West Ham
Chelsea have proven themselves to be a tough nut to crack during the Hiddink regime. On the flip side, West Ham will be confident and although they will be looking for a victory, a point would not be a disaster. This could be a game where the two teams cancel each other out.