Illinois Section 8 -- Tenant staying past lease date

Hi Guys,

I have a Section 8 tenant who I fear may over stay past their lease. I'm looking for contingency strategies on how to manage the situation.

But first the background:
- Acquired 3 Flat Building in Cook County  Feb 2016
- Inherited Section 8 Tenant with lease expiration June 30, 2016
- Rents are deposited via Direct Deposit by the Housing Authority of Cook County
- Rent is well below market, I'd say by ~$500 / month 
- Provided 90 day notice to tenant that I would not be renewing lease
- Tenant acknowledged and agreed to look for new apartment
- 2 weeks ago, tenant said they haven't found a new apartment yet and requested to stay longer. I informed that it wouldn't be possible as renovations are set to begin the week of July 5th.

Under normal circumstances, I would not accept any rent from a tenant who has over-stayed. Doing so compromises the eviction process. However, since the funds are direct deposited I'm  not sure what to do.

I'm guessing I'm not the first to encounter this problem -- Appreciate any words of wisdom.

Best,

Lou