The Mail on Sunday says there have been no talks with current boss Roberto Martinez about making the move to Merseyside permanent when his City contract ends this summer.
While there is a lot of middle ground to work out, Barry has made it clear he is open to staying at Goodison.
"If the club offer me a contract and show me they want me, then it is a possibility," he said.
"It won’t be a case of waving to the fans on Sunday. That would be like saying goodbye before I’ve made any decision. I’m just desperate to finish the season on a high.
"Roberto has been brilliant. I’ve learned so much from him, he’s got great ideas all the time that are ahead of the game. Even at 33, I’m still learning tactical stuff from him."
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