The club's Italian owner says he will know for sure on Sunday. Presumably this is when Milanic should have sorted out his severance deal with his current club, Sturm Graz.
Privately, Cellino admits he made an error in appointing Dave Hockaday - too inexperienced a manager and lacking the profile and probably the personality for such a big role.
He realised he needed a more experienced boss to come in, which inevitably meant spending more money.
The search led him to the 46-year-old Slovenian, who was an international player of some standing, and who has had success as a manager with Maribor in Slovenia, and then with Austrian side Sturm Graz since the summer of 2013.
And this might sound flippant, but it really isn't - Milanic is also not a bad looking bloke. This will not have escaped Cellino's attention.
He regularly commented on how ugly Hockaday was and even insisted this had held him back in his coaching/ managerial career.
There were other candidates for the job, of course. Cellino tried to get Oscar Garcia, and thought he had succeeded, only for Watford to beat him to the signature of the 41-year-old Spaniard.
He also met former West Brom manager Steve Clarke, but wasn't convinced the Scot really wanted the job.
In fact Clarke admitted he wanted to manage a Premier League side.
Cellino joked that Clarke had once been assistant to Gianfranco Zola (at West Ham), so can't have been taught very well.
The two Italians famously fell out when Cellino owned Cagliari and took Zola there as a player.
The 58-year-old also met former Leeds boss Simon Grayson - and his agent. The eccentric Italian joked that he wasn't sure who was the manager and who was the agent, because the latter spoke more than the former.
Cellino admits he's baffled and frustrated that managers (or head coaches) in England all seem to have agents, and that they exert so much influence.
He insists things are different in Italy.
Anyway, it looks like Milanic is the man, so long as he can sort out the final details of his severance with Sturm Graz, and we should know on Sunday.
Another chapter begins at Elland Road, with the club about to appoint their first foreign coach.
Cellino insists that Milanic will have a talented and improving squad of players to work with if he comes in. The Italian has been impressed with their recent progress under caretaker boss Neil Redfearn.