The Gunners are thought to be weighing up a £6million move for the nine-cap England international.
However, the ex-Norwich City shot-stopper maintains he is in no rush to leave and is happy staying at Upton Park.
Green is anxious to avoid another season of struggle and is backing joint-owner David Gold and David Sullivan to get things right off the pitch.
West Ham are on the lookout for a new manager following Gianfranco Zola's sacking, although Green is simply focusing on doing his best for the club rather than other issues.
"I have always stated that I am a happy Hammer," he told the Romford Recorder. "I want to be here and not fighting relegation.
"Realistically that's something that's out of my hands and they (the owners) can say as much as they want to say about it.
"I am sure the two chairmen will take their chance now to decide where they want to take the club and who that might involve.
"There has been a lot of talk and a lot of speculation and I am sure come the start of pre-season, things will be a lot clearer, but that is down to them.
"For me, my job is playing in goal and that is what I focus on. I am a bit too old and long in the tooth to get too focused on things that are out of my control.
"I just sit there and accept what comes along, focus on the 90 minutes every week when it comes along. That is my job and that is what I will carry on doing."