Could the Swedish superstar end up at the Olympic Stadium?
So should they take a punt on trying to bring one of world football's most enigmatic talents to the club?
The player in question is Zlatan Ibrahimovic.
The Swedish veteran hinted that he was open to a potential Premier League move for next season, as reported by Sky Sports.
So why should West Ham go for him and why is it not as unlikely as it might initially seem?
Here are three reasons.
The stadium move
West Ham's move to the Olympic Stadium is undoubtedly a big draw. The massive project looks set to put West Ham into the English elite and their fans should fill the impressive stadium.
They need a big name to go with that move and Zlatan would be the perfect eye-catching superstar to get pulses racing.
Big fish syndrome
At this stage of his career Zlatan has no interest in being a bit-part player. He will want to be the big fish at whatever club he ends up at - as he currently is at Paris Saint-Germain.
He would be the biggest earner and most famous star in the West Ham team if he did make the move and the fact the club's current striker spends more time on the treatment table then in the starting XI first-team football would be likely.
Where else in England could he go?
Manchester City and Manchester United appear to be no-go options. He has a strained relationship with both Pep Guardiola and Louis van Gaal - although, admittedly, LvG may not be at Old Trafford next season.
His relationship with Arsene Wenger is not much better. The Arsenal gaffer infamously failed to sign him as a teenager. He would not fit the philosophy of either Jurgen Klopp or Mauricio Pochettino at Liverpool and Spurs respectively.
That leaves Chelsea, but he criticised a bulk of their players openly after last season's Champions League clash between the sides, Sky Sports reporting he dubbed them 'babies'.
Outside of the usual suspects West Ham looks the best fit.
A new stadium, London-based and a mercurial manager who would allow him the freedom to be himself.
Zlatan to West Ham? Surely it could happen.