The 36-year-old had two spells as a player at the Toffees, coming through the youth ranks before returning towards the end of his career.
Rooney said he was aware of speculation linking him with replacing Rafael Benitez, but added there had been no official approach.
"Of course Everton is a club I grew up supporting," he said.
Rooney, who joined Everton as a nine-year-old and made his first-team debut at 16, has impressed managing Derby in difficult circumstances.
The Championship club were docked 21 points this season for entering administration and breaching financial regulations but Rooney has given them a chance of survival. An impressive run of form has moved them to within eight points of escaping the relegation places.
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