Lampard was dismissed on Monday with the Toffees in a run of bad form.
Goal reports the Premier League side have now made informal contact with Rooney.
Rooney is currently in charge of DC United in the MLS, but has serious reservations about returning to England and taking over at Goodison Park.
The former Manchester United striker has previously revealed he turned down the opportunity to manage Everton in January 2022 before Lampard was appointed.
“Everton approached my agent and asked me to interview for the job,” he said at the time.
“But I turned it down. I believe I will be a Premier League manager and I am ready for that 100 per cent. But I have a job at Derby, which is important to me.”