The hosts have not won in five - a run consisting of four defeats - while the Baggies are without a victory in eight as they continue to slide down the Premier League table
Downing's men have been in dismal form of late, picking up just a point from their last five top-flight matches and winning only once on the road in the Premier League this season.
Their most recent fixture saw them hold Tottenham 1-1 at White Hart Lane, but they will go into the final game of the calendar year in 15th place - a point off the drop zone.
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Just three points separate the two teams, with the London club failing to win in the league throughout December - their most recent defeat coming against Arsenal on Thursday.
Allardyce will hope for improvements in his forward line having averaged less than a goal a game in the top flight this term, although they remain without club-record signing Andy Carroll.
His absence has played a big part in West Ham's struggles, although fellow striker Carlton Cole has scored in his last two outings.
West Brom, who lost their first three games of the month, were able to stop the rot with a last-gasp draw against Hull City and followed that up with a hard-fought point against Tottenham on Boxing Day.
They will likely fall further into the relegation mire with defeat in east London but, based on the sides' recent head-to-head record, a point looks the most likely outcome on Saturday.
Four of the last five fixtures between the pair have ended in a stalemate, with two of those goalless.
Victor Anichebe and Nicolas Anelka are both suffering with groin injuries and are unlikely to feature for the visitors while Allardyce's only fresh injury concern is James Collins, who limped out of the 3-1 defeat to Arsenal.