It was not until after the summer window had closed at 11pm on Monday that the Toffees confirmed the deal for the 32-year-old midfielder.
And asked at a press conference on Tuesday if he had been worried the move would not go through, Barry said: "Yes.
"I think when you get into that last hour and there are still a few snags between the two clubs, with not everything sorted yet, you do fear the worst and start thinking you might be going into training with Manchester City.
"But I'm grateful it all went through."
Barry's contract with City, who he joined from Aston Villa in 2009, comes to an end next summer and he had been told by the Etihad Stadium outfit that he could not be guaranteed first-team football in the 2013-14 campaign.
He said: "It was important for me to make that decision - whether I was happy to see out my last year at Manchester City and not really feature at all.
"That was never something I was going to be comfortable with I think.
"I have been lucky enough to play regularly throughout my career.
"So I want to try to establish myself in the first team here, get a regular spot in the team and try to get a consistent run of games and good form together.
Source: PA
Source: PA