The striker's future has been uncertain since he was dropped by United manager Jose Mourinho last month, with possible moves to the United States and China being touted.
A return to his boyhood club which he left for United for 27 million pounds ($32.87 million) in 2004 would offer the 31-year-old Rooney a way to stay in the Premier League without even moving from his Cheshire home.
Asked on Friday if he would like to sign Rooney, Koeman said: "He is a great player and he still hasn’t finished his career. I don’t know what his situation is, I need to respect that.
"But even if there is a possibility that Rooney was an option for Everton, I would be very pleased."
Rooney's dealings with Everton have been strained, with some supporters critical of the way he left, but he has also been linked with a return by his former Everton team mate Leon Osman.
"I think he has been mending bridges for a couple of seasons now and you know he is an Evertonian at heart," Osman said.
"I am pretty sure he would love to come back and I hope it happens."
Rooney has not started a league game for United since facing Watford on Sept. 18 and missed last weekend's rout by Chelsea with injury.
(Reporting by Neil Robinson; editing by Rex Gowar)
