The centre-back has emerged as one of the finest defenders in the British Isles, and manager Sam Allardyce is hoping that he can do a deal for the Dutchman in January.
Reports had previously linked Arsenal with making a £7 million move for the 23-year-old in the next transfer window, but it seems that the Hammers could be set to rival them.
Celtic's failure to qualify for the Champions League this season is thought to have been the decisive factor behind them being resigned to losing Van Dijk.
The Bhoys lost a lot of money from their early exit in qualifying and it also remains harder to keep the player at Celtic Park without the prospect of playing elite level European football.
Of course, West Ham also cannot offer this, but the Hammers' start to the season has been impressive. At this moment in time they are ahead of Arsenal, and they will be hoping to continue this form to ensure they are an attractive move for Van Dijk.
The club are in danger of losing Winston Reid and, althoughAllardyce would ideally like to partner the pair, Van Dijk could make a good replacement for the New Zealand international.
He also fits in with their new-found passing style of play, which has served them so well so far. Allardyce's side have placed a lot more importance on retaining possession this campaign and Van Dijk would certainly be suited to a free-flowing game.
He has the technical skill to match his defensive attributes and would be an upgrade on Allardyce's current options if he was to make the move to Boleyn Ground.