Released Newcastle United coach Steve Round is keen on a move to Everton.
Everton manager Moyes interrupted his holiday to call Newcastle first team coach Round and ask him to fill the role vacated by Alan Irvine last November.
He has been given permission to speak to the club as Moyes looks to install his new assistant ahead of the return to pre-season training next Thursday, after Round's current contract at St James's Park expires on July 1.
But Chelsea and Twente - who Round's former colleague Steve McClaren took charge of earlier this week - are also in the hunt for his services.