The Toffees boss Moyes has yet to commit himself to Goodison Park past the end of the season and is now on Chelsea's shortlist of new managers.
But the Scot may be tempted to see if the City job becomes available before making any decisions about his future. Manchester United have ruled out a move for him as he has virtually no Champions League experience.
Chelsea owner Roman Abramovich has taken soundings from players on his next appointment and Moyes features highly on the list of candidates, with Malaga's Manuel Pellegrini and Borussia Dortmund's Jurgen Klopp. Jose Mourinho is also pushing his case for a return.