Tex. Finance Code § 36.105

(a) A contract between the bank and a person for bailment, of deposit for hire, or for lease of a safe, vault, or box ceases on the date specified in the notice as the date for removal of property or a later date approved by the banking commissioner. A person who has paid rental or storage charges for a period extending beyond the date designated for removal of property has an unsecured claim against the bank for a refund of the unearned amount paid.

(b) If the property is not removed by the date the contract ceases, an officer of the bank shall inventory the property. In making the inventory the officer may open a safe, vault, or box, or any package, parcel, or receptacle, in the custody or possession of the bank. The inventory must be made in the presence of a notary public who is not an officer or employee of the bank and who is bonded in an amount and by sureties approved by the banking commissioner. 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 bank's custody and control has been inventoried, a master list certified by the bank officer and the notary public shall be furnished to the banking commissioner. The master list shall be kept in a place and dealt with in a manner the banking commissioner specifies pending delivery of the property to its owner or to the comptroller as unclaimed property.