Sky Sports says Rooney, who made his Premier League debut for Everton at the age of 16, admires his former club and would consider playing for them again after falling down the pecking order at Old Trafford.
Rooney is understood to want to see out the remainder of his United contract, which runs until the summer of 2019, but the lack of playing time under Jose Mourinho has forced him to assess his options.
His agent Paul Stretford flew to China last month to hold talks over a potential move for the England striker to the Chinese Super League.
Everton manager Ronald Koeman confirmed this week that he was interested in bringing Rooney, 31, to Goodison Park in the summer.
"He still has two or three years in front of him to play at the highest level," the Dutchman told Sky Sports.
"What will happen at the end of the season, I do not know. In my opinion he's one of the players who can make Everton stronger than they are now.
"I still think Wayne Rooney is playing at a high level and he made a good choice staying at Manchester United and in the Premier League.
"It's all about what the player likes, what Manchester United need to do and we are not involved in that project. Every player who we feel can make the team stronger is welcome at Everton."
Everton are reported to have approached United in January to enquire about the possibility of signing Rooney on loan for the rest of the season.
But Old Trafford bosses immediately rebuffed the approach, insisting that the England captain remained a key part of their plans.
Rooney has made 17 league appearances for United this season, scoring two goals. He overtook Sir Bobby Charlton to become the club's all-time record goalscorer when he netted an injury-time equaliser at Stoke in January.