Tex. Finance Code § 36.222

(a) A contract between the bank and another person for bailment, of deposit for hire, or for lease of a safe, vault, or box ceases on the date specified for removal of property in the notices that were published and mailed or a later date approved by the receiver or the court. A person who has paid rental or storage charges for a period extending beyond the date designated for removal of property has a claim against the bank estate for a refund of the unearned amount paid.

(b) If the property is not removed by the date the contract ceases, the receiver shall inventory the property. In making the inventory the receiver may open a safe, vault, or box, or any package, parcel, or receptacle, in the custody or possession of the receiver. The property shall be marked to identify, to the extent possible, its owner or the person who left it with the bank. After all property belonging to others that is in the receiver's custody and control has been inventoried, the receiver shall compile a master list that is divided for each office of the bank that received property that remains unclaimed. The receiver shall publish, in a newspaper of general circulation in each community in which the bank had an office that received property that remains unclaimed, the list and the names of the owners of the property as shown in the bank's records. The published notice must specify a procedure for claiming the property unless the court, on application of the receiver, approves an alternate procedure.