I wouldn't think it will come down to Monchi or nobody in the board's opinion. As with Koeman, I would imagine they have a shortlist of 4-5 people and would not prefer to make an appointment outside of that list.
It will be dependent on who is available and willing, as with everything else. I do hope this is something that gets priority and is being progressed, but I have the feeling that it is not going to happen in the short-term/next couple of months.
Koeman and others will be doing all of the groundwork in relation to incoming transfers for the summer and I imagine the DoF is seen as more of a strategic development, for which they will be afforded more time to get the right person in.
Kenwright's recent comments would also suggest he viewed himself as Quasi-DoF which was a little surprising, even though he is an experienced transfer negotiator, the role is far greater in scope and depth than that.
If Dein were to be appointed as Chairman and Kenwright's role reduced to goodwill only, it may be at that stage that they complete a more fundamental shake-up of the board and management, and appoint a DoF as part of that re-structuring.