Allardyce said his appetite for winning football matches is stronger than ever as he plots another Premier League survival bid.
Hegazi was unused substitute in West Brom's Premier League opening game against Leicester City.
Moyes, who replaced the sacked Slaven Bilic in November, guided the relegation-threatened Premier League club to a 13th-placed finish.
When Bilic was sacked and David Moyes was appointed as the new manager, the former Stoke City winger turned out to be the team’s main star.
Pearce experienced sharing the bench with Moyes in Preston North End and Everton.
Moyes has a great experience in the Premier League; he managed Everton, Manchester United and Sunderland.
Bilic, 49, wore the West Ham shirt for more than 45 games during his only season with the club in 1997.
West Ham United have sacked manager Slaven Bilic after they dropped into the relegation zone following Saturday's 4-1 home defeat by Liverpool.
West Ham suffers this season despite the high expectations from the team in the pre-season, as they are in 16th place