According to the Daily Star it looks like Everton are discussing Joe Royle as the future director of football. This position would be a connection between manager and the board and drive the organization. It seems like a position that will either be announced at the same time as the new manager or after. If Everton are somehow able to convince a manager like Jose Mourinho to come to the Toffees I don't expect any director to be hired.
So the team is unlikely to hire a director of football if they think they have any chance at a huge name like Mourinho. Jose is not going to want anyone else overseeing sales and acquisitions, which he would manage on his own with a big check required from the board to cover his deals.
Joe Royle was involved in the post Roberto Martinez interim for the final game and would be a solid choice for the position. He is still a fan favorite and was welcoming of the assistant job to David Unsworth for that final game. How he would approach the job and work in the modern game after not managing since 2009 is anyone's guess, but he would be a good choice.
On the other hand the manager front has turned back to Lucien Favre today in fresh rumors from a Swiss daily. The news says Everton have contacted his representatives so it's not very far if true. He's been linked for a while now so it's surprising if they are just contacting his representation now.
Favre is another solid and stable choice for the Everton team. He has a very good history in the Bundesliga with several squads. He is also out of work and would not require any compensation. How much compensation plays in Everton minds this time is unclear, but it wouldn't hurt to aquire a manager without having to raid the player transfer funds.