The Netherlands international has not featured in the top flight this season and is reportedly keen to leave Goodison Park in order to boost his chances of featuring at the 2014 FIFA World Cup.
Allardyce confirmed Roberto Martinez has agreed to let the 30-year-old move to Upton Park, with personal terms only standing between a deal being completed.
A spate of injuries has left West Ham heading into 2014 with no fit centre-backs. They ended Saturday's 3-3 draw with West Brom with a defensive comprised of four full-backs, George McCartney and Guy Demel moving centrally.
"We've had the OK from Everton and we're talking to his agent at the moment," Allardyce is quoted as saying by BBC Sport.
"So hopefully that will happen in the next day or so."
West Ham have also been heavily linked with a move for former Sunderland man Asamoah Gyan in January, although the Ghana striker has insisted he wishes to stay with his current club, Al Ain.
