Nilsson was given a chance to impress at the Boleyn Ground earlier this month and he has obviously done well during his brief showings on the training pitch.
Nilsson is currently a free agent and the Hammers reportedly believe that he could offer a bargain buy, who would add cover to their squad.
West Ham are starting to plan for life without Winston Reid, who is said to be set to leave for Arsenal or Tottenham Hotspur in the summer by the Daily Mail, and Nilsson may be lined up as his long-term replacement.
A one-time Swedish international, the commanding defender was tipped to have a bright future in the game, but his career has rather stalled in recent times.
Moves to FC Utrecht in Holland and Kalmar FF, back in his homeland, have not really worked out and he will be hoping that a switch to West Ham, if it is completed, will change his fortune.
If Nilsson does sign, he could join Darren Fletcher and Stephen Hendrie who the Daily Mail insist the London outfit are also trying to snap up before the end of the month.