Yes. Most all landlords and owners will check your background unless you have a very good credit rating and a large amount of liquid assets. With that said some will adjust their requirements depending on your situation. The first question I would ask is why are you leaving and getting out from under your current/past lease? Why not stay until the lease ends? I would also ask if you spoke with them about breaking the lease early and reducing the penalty you would pay. Realize that some landlords and owners are picky while others are not provided your situation can be explained rationally. I just placed someone in the military with bankruptcy on their credit but his situation was somewhat unique. Three agents he spoke with before me told him to forget about renting a house and looking at apartments. Long story short find advice from someone that actually works in the property mangement business, don't trust just any Realtor/Landlord/Owner.
We manage over 100 homes in Southwest Houston and have done so for over 15 years so if you need any help or advice feel free to contact us.