Allardyce revealed on New Year's Day that the London club were in talks with the Netherlands centre-back's agent after Everton gave them permission to speak to the 30-year-old.
West Ham were forced to change formation in their 2-1 defeat at Fulham and have no fit centre-backs after a hectic festive schedule.
Heitinga initially welcomed West Ham's interest in him, but the former Atletico Madrid man has opted against joining the Premier League strugglers.
He told De Telegraaf: "The deal has been cancelled. This has nothing to do with money, but is only about the feeling I got.
"This might be the most important decision of my career, so everything has to be just right. We reached the conclusion that this is not the case after we held talks."
Heitinga has not featured in the Premier League this season and appears to be surplus to requirements at Goodison Park.
Youngster John Stones featured in the heart of defence alongside Antolin Alcaraz in Wednesday's draw at Stoke City in the absence of first-choice pair Phil Jagielka and Sylvain Distin.
The combative defender has also been linked with a move to Serie A high-flyers Roma.
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