On one side of London, they are revelling in their most enjoyable season since Alan Pardew’s briefly endearing collection of young hopefuls, wily old pros and outsiders were denied their moment of glory by that improbable rocket of an equaliser from Steven Gerrard in the FA Cup final 10 years ago. Almost 12 miles to the west, meanwhile, they are struggling to absorb the implosion that has left the fallen champions wondering if they can muster the enthusiasm for a late push to qualify for the Europa League.

While West Ham United could yet find themselves toasting the implausible double of a place in the top four and their first FA Cup since Sir Trevor Brooking scored a rare headed goal 36 years ago, you have to go back to the 1992-93 season and the start of the Premier League to find the last time that Chelsea had nothing but pride to play for before the clocks went forward. They were a wildly different club back then, barely recognisable from the modern moneyed version, and bearing in mind how much they have lorded it over their East End neighbours over the past two decades, it is certainly a strange state of affairs that West Ham will back themselves to secure their first league double over Chelsea in 13 years when they visit Stamford Bridge on Saturday afternoon.

José Mourinho’s dismissal two months after Chelsea’s controversial 2-1 defeat at Upton Park in October could not disguise the structural flaws within the squad. Chelsea have not improved enough to threaten the top four since Guus Hiddink took caretaker charge; Diego Costa is suspended; they are in danger of grinding to a halt after being bumped out of the Champions League by Paris Saint-Germain and the FA Cup by Everton in the space of three days; and, barring a late surge in form, next season will be notable for the absence of any European nights at Stamford Bridge for the first time in 20 years.

No one was predicting at the start of the season that West Ham would be fifth with nine games left, yet Slaven Bilic’s side are nine points clear of 10th-placed Chelsea and two behind Manchester City in fourth, Dimitri Payet has been one of the signings of the season and they will have an FA Cup semi-final against Everton to look forward to if they win their replay against Manchester United at Upton Park next month.

Similar in size to Chelsea until the west Londoners began their transformation from mid-table lightweights into cosmopolitan entertainers capable of challenging for major honours in the mid-90s, West Ham have been the poor relations in a one-sided rivalry for a long time and their trips across the capital have mostly been occasions to forget since Roman Abramovich arrived in England in 2003. They have not finished above Chelsea since 1996 (10th to Chelsea’s 11th) and the closest they have got to them since was in 2002, when Glenn Roeder’s side finished seventh, one place and 11 points behind Claudio Ranieri’s men.

This Paolo Di Canio goal, scored past Carlo Cudicini, secured West Ham’s last league double over Chelsea, in 2002-03, a season which ended in relegation for the Hammers and a Champions League place for Chelsea. Photograph: Tom Jenkins/The Guardian

West Ham lacked ambition and vision during that period. Their academy produced a golden generation that could have rivalled Manchester United’s Class of 92, but it was Chelsea who signed World Cup winners and stars from Serie A and La Liga, Chelsea who ran a formidable Barcelona close in a brilliant Champions League quarter-final in 2000, Chelsea who won two FA Cups, a League Cup and the European Cup Winners’ Cup.

In the 2002-03 season, West Ham beat Chelsea home and away, but the victories were bittersweet. Paolo Di Canio cracked a spectacular volley past Carlo Cudicini in a 3-2 win at Stamford Bridge in late September, but it was West Ham’s first win in a harrowing season, and although the Italian also scored the winner over Chelsea in May, the clubs were heading in opposite directions.

A week later, as West Ham went down, Chelsea qualified for the Champions League with a win over Liverpool and it would soon transpire that their defeat at Upton Park was the penultimate match of the pre-Abramovich era.

A relegated West Ham were easy prey. Glen Johnson became the first player to sign for the new Chelsea and Joe Cole, West Ham’s captain, also made the journey across London. Frank Lampard had already joined for £11m two years earlier and whereas Chelsea have lifted four Premier League titles and become the first London team to win the European Cup in the 13 years since Abramovich’s takeover, winning 15 trophies overall under their Russian owner, in that time West Ham have accumulated three seasons in the Championship, bounced back from relegation twice, made enemies during the Tevez and Mascherano fiasco and flirted with bankruptcy under Icelandic ownership.

Frank Lampard, left, and Joe Cole joined Chelsea from West Ham and Glen Johnson made the same journey in a sign of the balance of power. Photograph: Alex Morton/Action Images

A little over five years since they were bought by David Gold and David Sullivan, however, West Ham are in their strongest position since finishing fifth under Harry Redknapp in 1999 and hope that this summer’s move to the Olympic Stadium will give them a better chance of competing with Arsenal, Chelsea and Tottenham Hotspur.

It is a small sample size and West Ham should not kid themselves into thinking that they have overtaken Chelsea on the basis of one weird season. Chelsea, with Antonio Conte likely to become their new manager when Hiddink steps aside, will expect to recover from a blip that can be attributed to Mourinho’s loss of control and a recruitment strategy that has left much to be desired.

All the same West Ham, energised by Bilic’s relentless positivity, have taken several scalps this season and after years of going to Stamford Bridge with nothing more than damage limitation on the mind, they can finally look Chelsea in the eye.