The Sunday Mirror says top clubs have been impressed by the way Mackay has guided Cardiff to the Premier League for the first time in 51 years.
Mackay is almost a carbon copy of Moyes. Both Scots, they share the same hard-graft, no-nonsense qualities.
Moyes cut his managerial teeth at Preston North End before joining the Toffees 11 years ago.
His current contract expires this summer and there is a growing belief inside the game that he will choose to move on.