The minimum amount of time is about 65 days of delinquency. However, the big banks are so inundated with loans in default that would not be a reasonable estimate. Some homes have not had a single payment in 2 years and there is still no notice of sale (foreclosure).
In addtion to each bank having its own policies, territories are often divided up and managed by different people. Since some states take a year-plus to go from filing to foreclose to the actual proceeding, the loans in default are handled differently there. So, while Texas has the same laws, the banks are all different and the asset managers are, too.
If you ask about a particular address in public or private, we'll try to help you.
With short sale transactions, you need a Realtor who has information and knowledge.
You are free to contact me to assist you.
Real Estate Broker
Omega One Realty
Each house is short sold/foreclosed on a case by case basis. There is no 'rule of thumb' regarding a timeline since each borrower/buyer/lender situation is different. I've seen some homes list as short sales for a month then go into foreclosure and other times stay on the market for a year.
Good luck in your search.
PS. it works that way across Texas (not just Dallas)