And boss Glenn Roeder says he was "not able to take it in" when he first heard the news.
Les, who left the club this summer after almost six years service as a player and coach, died after suffering a heart attack on Sunday.
He leaves a wife, Elaine, and two teenage sons, Joe and George, who are both on the club's books and, like Les, are goalkeepers.
Says Paul: "On behalf of everyone at the football club I would like to express our sincere condolences to the family of Les.
"He had a long and illustrious playing career, which culminated in him joining West Ham.
"When he made his debut with us, he said it was the fulfilment of a lifelong dream.
"His subsequent work for us as a goalkeeping coach and providing assistance on the bench during games was performed with great energy and enthusiasm.
"Although he left the club in the summer, the family connection has continued through Joe and George, and we will help them in whatever way we can to come to terms with this tragic loss.
"Les was a larger than life character who will be sorely missed by everyone who knew him, and everyone's thoughts at the club are with his family at this difficult time.
Manager Glenn adds: "Roger Cross rang me to tell me the sad and unbelievable news, and I was not able to take it in.
"You do not expect anyone in their 40s to die of a heart attack, and it is awful.
"It is difficult to know what words can express our sadness other than to say that our thoughts and sympathies are with his wife Elaine and the family, including Joe and George, his sons who are trainees at the club.
"I hope everyone will respect their privacy after this tragic news."
Although his playing career took him around the country, most notably to Manchester United, against whom he made a brief substitute appearance in West Ham colours at Old Trafford in May 1997, Les will be best remembered by Hammers fans for his outstanding performance up at Newcastle in a 3-0 defeat, and coming on as an outfield player against Arsenal at Highbury.
He said his only regret in football was not actually playing at Upton Park for the Hammers.
His career lasted 22 years and took in 564 games. He also played for Coventry City, Blackpool, Birmingham City, Aston Villa, Plymouth Argyle, Luton Town, Bury and Leyton Orient.