We study modules with the properties (delta-TWE) and (delta-TWEE) which are adopted Zoschinger's modules with the properties (E) and (EE). We call a module (delta-TWE) module if M has a weak delta-supplement in every torsion extension. Similarly if M has ample weak delta-supplements in every torsion extension then M is called (delta-TWEE) module. We obtain various properties of these modules. We will show that (1) Every direct summand of a (delta-TWE) module is a (delta-TWE) module. (2) A module M has the property (delta-TWEE) iff every submodule of M has the property (delta-TWE). (3) Any factor module of a (delta-TWE) module is a (delta-TWE) module under a special condition. (4) Over a non local ring, if every submodule of a module M is a (delta-TWE) module, then it is cofinitely weak delta-supplemented.