With Jonathan Woodgate and Ledley King suffering injury problems, Tottenham manager Harry Redknapp sees Upson as the perfect choice to help Spurs continue their battle for a top-four finish.
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West Ham could also lose the service of Scott Parker, who also believed to being monitored by Redknapp.
Spurs, meanwhile, have been linked in France with a move for highly-rated Bordeaux striker Marouane Chamakh.
Chamakh's contract expires in the summer and he looks set to move in January with Arsenal remaining keen on the forward, particularly as Robin van Persie is out for most of the season.
The reports claim tentative discussions have already taken place between officials connected to the striker and Spurs, but that Everton and Sunderland are also keen.