West Ham said on their website on Thursday that he suffered a "deep laceration" of the toe after he trod on a piece of the broken cup on Tuesday.
"The resulting wound required emergency hospital treatment to clean and stitch the wound," said the Hammers on their website with manager Sam Allardyce later saying he would probably miss Saturday's Premier League clash with Arsenal.
"We're not exactly certain but we think it will be too soon for him be considered for the game," Allardyce said at his pre-match briefing.
"It was a freak accident at a time when we could have done without it due to our lack of available strikers."
Valencia, who scored all three of his country's goals at last year's World Cup, has made 30 appearances and scored five times since joining the Hammers from Mexican side Pachuca.
West Ham have slipped down the table after scoring only four goals during a winless run of seven matches and are also without striker Andy Carroll who is out with a knee problem.
Valencia is not the first footballer to miss a match because of a domestic accident.
Former England forward Darius Vassell once infected a blister on his toe after he used an electric drill to try and drain it, while former Wimbledon keeper Dave Beasant damaged himself by dropping a jar of mayonnaise on his ankle.
Alan Mullery missed out on making his debut for England in May 1964 after suffering a muscle injury while having a shave and had to wait another seven months to play for his country for the first time.
(Reporting by Mike Collett; editing by Martyn Herman)
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