Manchester City have also been linked with the 23-year-old in recent seasons but Dunne, who plied his trade with the Citizens between 2000 and 2009, thinks Barkley would be wise to move to north London.
Everton are setting their sights on transfer window targets as they gear up to meet the demands of European football next season, manager Ronald Koeman said on Friday. If he stays there will be more competition for him next season.
Ronald Koeman admits he doesn't feel confident that Ross Barkley will sign a new contract at Everton and that he expects the situation to be resolved by Monday.
"I am 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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The 36-year-old defender, who will leave Chelsea this summer after 22 years, made his first Premier League start since September and 716th Blues appearance in last Monday's win over Watford .
"We will see what the answer will be - and I don't know what will happen if he doesn't give any answer about that".
Koeman is relaxed and despite wanting Barkley to stay confirmed the club are pondering other options.
Former Arsenal star Paul Merson has said that Ross Barkley would struggle to gain game time at Arsenal or Tottenham. "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".
Koeman's men face a trip to Arsenal on Sunday to complete their Premier League campaign and, with Barkley yet to sign the deal offered by Everton, the Dutchman signalled that the midfielder could now leave.
"What we need is to keep his goals and find more players to score more than four or five in one season".
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