The 26-year-old has gradually fallen down the pecking order at West Ham, with the likes of Joe Cole, Matt Jarvis and Stewart Downing having joined the Hammers in the last 12 months.
Vaz Te scored the winning goal in the Championship play-off final that sealed West Ham's return to the top flight and also impressed during their first season back in the Barclays Premier League last season.
A dislocated shoulder in the 3-1 home defeat to Arsenal curtailed Vaz Te's season but the player, who also worked under Allardyce at Bolton, has said he is disappointed with his situation at West Ham.
"Last season I was asked to change positions for the benefit of the team even though I had just scored 24 goals. Despite this, it has been made obvious to myself and my representatives that I am not in the manager's plans," his statement, issued on Thursday, read.
"In my 10 years as a professional football player I have never submitted a transfer request nor issued a statement. However, I am frustrated at West Ham United Football Club."
The timing of the statement, with just a matter of days remaining of the summer transfer window, has left Allardyce questioning the reasoning behind making such comments public.
"If anybody's going to leave now it'll be Vaz," he said. "You don't keep anybody that doesn't want to stay. If a club comes in to make the right offer for him we'll be more than happy to accommodate the transfer request he's given us.
"Generally when you make [a statement] like that, they've been tapped up. Someone would've probably already contacted him.
"I am expecting a bid for him in the next couple of days because you don't do that sort of thing this late in the transfer window.