Out-of-favour Manchester United striker would be welcomed back at Goodison Park by his boyhood club after being linked with a move
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Rooney is currently out of favour at Manchester United with reports suggesting he has been told by Jose Mourinho that he can leave Old Trafford.
The England captain has been linked with a move to Los Angeles Galaxy to join up with fellow Merseysider Steven Gerrard but Koeman would relish the opportunity to bring him back to his boyhood club.
Speaking ahead of Sunday's clash with West Ham, the Dutchman told reporters at his news conference: "First of all, he's a great player - he's still not finished his career.
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"I don't know what his situation is, I need to respect his situation. That's not my problem.
"Even if one time there is a possibility that Rooney is an option for Everton, I'd say 'yes, please'."
On his struggles at United, Koeman added: "That's football.
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"He is enough of an experienced player and person that he knows what can happen.
"Everyone gets periods of criticism. It's part of the business and you need to stay calm and that's the best thing you can do."
Koeman's willingness to discuss Rooney appears to be part of a charm offensive initiated by Everton midfielder Leon Osman, who earlier on Friday told Sky Sports: "[A return] would not surprise me.
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"I think he has been mending bridges for a couple of seasons now and you know he is an Evertonian at heart.
"I am pretty sure he would love to come back and I hope it happens."
Rooney started his career at Everton, scoring a wonder goal against Arsenal aged just 16. He joined United in August 2004 and has been a regular for club and country ever since until this season, when he has struggled for playing time under Mourinho.