Overstaying at end of tenancy

I am wondering what our position is with some tenants on a periodic tenancy?
They gave just over one calendar months notice of their intention to move out, and asked of they could leave mid-month possibly staying more, but were vague about how long and wanted to stay week-by-week, thus giving us no notice. We sent them notification when they sent their intention to leave that they could leave mid-month and gave a specific date to terminate with pro-rata adjustment for the rent. However when I have sent them notification of the check-out date (with 2 weeks to go) they have said they want to stay till the end of the month maybe more (they are willing to pay rent), but I have renovation work scheduled and new tenants lined up. Where do we stand with this.

Thanks

Personally I would not have scheduled any renovation till they moved out .But its easy for me to say that, as I am a builder. I dont advertise till a tenant has left

The decision rests with you. If you have new tenants lined up and renovation works planned you must make this clear to your current tenants in my view and come to an agreed end-date.