Diame, 25, only joined the Hammers in the summer from Wigan, but is understood to have a £3.5mi buy-out clause. While there has been no official approach lodged by the Gunners as yet, they are reported to have scouted Diame in West Ham's midweek FA Cup defeat at Manchester United.
Allardyce accepts the situation is difficult, saying: "The owners have tried to renegotiate the contract, but they haven't been successful at the moment."
West Ham have been active in the January transfer window themselves, signing Joe Cole from Liverpool, while sorting loan deals for forwards Marouane Chamakh, from Arsenal, and Brazilian Wellington Paulista.
With England striker Andy Carroll set to return from injury sooner rather than later, Allardyce played down suggestions he would move for veteran Frenchman Nicolas Anelka.
"At the moment I am not in a position for Nicolas Anelka because my main priority is defenders," said the Irons boss. "We have got Andy close to coming back and we have got Carlton Cole, Chamakh and we have also brought in Wellington."
Allardyce hopes the 28-year-old from Cruzeiro will soon prove a valuable addition to his squad.
"We are not quite sure whether he will be able to go straight in, but he is a talented forward from South America and we hope he will be successful," said the Irons boss, whose side host Premier League relegation battlers QPR on Saturday.
"We have not had a lot of time to see him yet because of the travelling and games, but we have seen the usual raw talent of a South American player. The interesting part is whether or not he can integrate into the way the game is played in England and produce that skill.
"There will be an introduction at some stage and I think starting off the bench somewhere along the line in the near future. Who knows how long he will take to be ready? I could toss a coin on that one. Some can come straight in and do it and some never do it at all, some do it a bit later on."