The midfielder was available on a free transfer after coming to the end of his contract at Manchester City, but chose to head back to the Toffees after a loan spell there last season.
The 33-year-old told The Telegraph: "America was an option and there was a conversation about it but when I spoke to the coaches here they said, 'If you are going there, you are retiring'.
"I am sure I would have enjoyed it and there was a strong argument to go, but with the season we had at Everton last year I decided I was not ready to move and the club offered me a three-year contract which showed the faith they had in me.
"There was interest from three or four Premier League teams and a couple from abroad but I was happy here. While I could still play in the Premier League I really did not want to go abroad. I was not thinking about winding down my career."
Barry made 32 league appearances for Everton last season, and has featured in every Premier League game so far this campaign.