Tenants Must Prove they are moving
I have seen many landlords over the years eat another months rent by listening to the tenant saying I am moving at the end of the month and the Landlord’s wishful hoping they do. But guess what 80% of them DON’T.
You should ask for proof that he will be out. Like a signed lease agreement with another Landlord. If he cannot provide you with any documentation of proof then I would proceed with the eviction. My experience has been the tenants BS all the time saying I am going to move and then never do. You have to weigh the cost if he stays then how much more will it cost you compared to making sure and starting the eviction process now, because I just do not trust these tenants.
If they are behind in rent and you have believe their stories up until now… what makes you think they are telling you the truth now?