West Ham's joint chairmen David Gold and David Sullivan, chief executive Karren Brady, Allardyce, head of recruitment Martin Glover and club coaching staff started getting offered players once the clock hits midnight on December 31.
Allardyce explained: "I will be able to tell you exactly how many players West Ham have been offered when the window closes at the end of the month because I keep a note of everything.
"Right now I can tell you that it will run into the thousands but the window has always been out of control since it started 10 years ago. It has never been in control.
"It has never been an ideal situation in any way, shape or form during the season.
"It's obviously a very disruptive time of the year, the back end of December and the beginning of January.
"I try very hard not to let the window use up all my time - I pass it all on to Martin - and he is panicking a bit at the moment because he can't get any sleep at all."