The Toffees face Arsenal at the Emirates in their final match of the season on Sunday knowing that they will finish seventh in the Premier League.
"Maybe an Arsenal or Tottenham will be better for him [than Manchester City]", added Dunne, who now works as a pundit.
Ronald Koeman says he is not confident Ross Barkley will stay at Everton and has begun the search for a midfield replacement.
When asked if he feels confident that Barkley could stay, Koeman responded "No", before continuing: "It's a long time that he's thinking about his future but OK, I'm not anxious because I like to work with players who like to stay".
"If we go on holiday we need to know".
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Whichever player Koeman chooses to chase in the summer will give a strong indication of the style of play he wants to implement at Everton next term. I don't know what will happen after it.
Everton are said to be considering offers for Alex Oxlade-Chamberlain of Arsenal and Swansea City's Gylfi Sigurdsson should Ross Barkley decide to leave the club, according to The Sun. "I don't need to explain more because if we don't want the player to stay longer, then we don't offer him a new contract".
"No, because it's a long time that he is thinking about his future", Koeman said. He knows he can improve in several aspects, to keep the ball better, be stronger and do better pressing.
"What we need is to keep his goals and find more players to score more than four or five in one season".