Having been released by Besiktas, Oxlade-Chamberlain began training with Arsenal's U21 team this week.
Now Oxlade-Chamberlain has revealed he has turned down several clubs and is waiting for the right opportunity in England.
"Yes, but to be fair most (bids) have been to Europe, away from home. I spent two years abroad, which I found difficult at family level," he said on Fozcast.
"I went six or eight weeks without seeing my son and my wife. It was hard, and I knew I didn't want to do it again - I wanted to be back in England. It's just about waiting for the right thing to appear here, it makes sense for my family.
"Once you have kids, things change - it's not just about me now, or about football, it has to suit them too. I have received offers, and I have had to say no to them because they did not suit me and us. It's about waiting for the right project, something I can be enthusiastic about, getting into and getting started right away."