Adrian was only recently given a two-year contract by the East London outfit, but the Spaniard has been limited to two appearances in the EFL Cup this season.
Manchester City loanee Joe Hart is now regarded as the first-choice stopper by Slaven Bilic, and Adrian has acknowledged that he wants to be allowed to leave the London Stadium in order to gain regular action.
The 30-year-old told Radio Marca: "The coach is not giving me much ball this year, but the data is there. It only remains to fight. The possibility also to play for the national team when you are not playing also disappears.
"The market will be reopened and an agreement must be reached because I am not happy in this situation. Try other leagues? I would not mind. I want to enjoy every Sunday and that is achieved by playing."
Adrian has made a total of 125 appearances for West Ham since leaving Real Betis in June 2013.