Yes. If the buyer in this situation chooses to request an extension of the termination-option period instead of exercising the default remedies available to him in the contract, then he must agree to offer something of value as consideration to the seller to ensure that the extension is legally enforceable. This is often done by paying an additional termination-option fee.
Extensive case law in Texas suggests a termination-option period cannot be extended without an additional option fee, so a buyer should pay another option fee to reliably extend the option period.