Moyes is being linked with Sir Alex Ferguson's job after the Scot announced his retirement.
The 50-year-old's Everton contract runs out in the summer and no agreement has been reached over a new deal.
He has previously said that he wants to wait until the end of this campaign before deciding whether to extend his time at Goodison Park into a 12th year.
Moyes has brought stability and consistency on a relatively small budget to the Merseysiders and is viewed by United chiefs as a man who, like his outgoing compatriot, values youth development and longevity.
Everton have yet to comment on the speculation.